Hypothesis Tests for Population Means...logo1 Testing Normal DistributionsExampleSample Size Determination t-tests Hypothesis Tests for Population Means Bernd Schroder¨ Bernd Schroder¨

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Testing Normal Distributions Example Sample Size Determination t-tests

Hypothesis Tests for Population Means

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Introduction

1. Once the idea of hypothesis tests is understood, we want toset up standard procedures for frequently used tests.

2. The setup is always a test to reject a given null hypothesisH0 in favor of an alternative hypothesis Ha.

3. The probability α of a type I error (the significance level ofthe test) will be given before the test.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Introduction1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.

2. The setup is always a test to reject a given null hypothesisH0 in favor of an alternative hypothesis Ha.

3. The probability α of a type I error (the significance level ofthe test) will be given before the test.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Introduction1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.

3. The probability α of a type I error (the significance level ofthe test) will be given before the test.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Introduction1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.3. The probability α of a type I error

(the significance level ofthe test) will be given before the test.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Introduction1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.3. The probability α of a type I error (the significance level of

the test)

will be given before the test.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Introduction1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.3. The probability α of a type I error (the significance level of

the test) will be given before the test.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ .

For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0

,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)

= P(

X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)

= P(

Z >c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)

zα =c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

n

is the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . For H0 : µ = µ0 and Ha : µ > µ0,the rejection region should be an interval (c,∞) for somec > µ0.

α = P(X > c

)= P

(X−µ0

σ/√

n>

c−µ0

σ/√

n

)= P

(Z >

c−µ0

σ/√

n

)zα =

c−µ0

σ/√

nis the cutoff point.

For Ha : µ < µ0 or Ha : µ 6= µ0 computations are similar.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ .

To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses

, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:

Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

-

µ0

α

Upper tail test for µ≤µ0:Tail probability is ≤α (small) if µ≤µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)

2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:

Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

-

µ0

α

Lower tail test for µ≥µ0:Tail probability is ≤α (small) if µ≥µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)

3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2

α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2

α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:

Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population withknown standard deviation σ . To test the null hypothesis µ = µ0against various alternative hypotheses, we use the test statistic

z =x−µ0

σ/√

nand define the following rejection regions.

1. For Ha : µ > µ0 use z≥ zα (upper tailed test)2. For Ha : µ < µ0 use z≤−zα (lower tailed test)3. For Ha : µ 6= µ0 use |z| ≥ z α

2(two-tailed test)

-

µ0

α

2α

2

Two tailed test for µ 6= µ0:Tail probability is α (small) if µ=µ0.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

1. Determine the parameter of interest.2. Determine the null hypothesis H0.3. Determine the alternative hypothesis Ha.4. Choose the appropriate test statistic.5. Determine the rejection region using the significance level.6. Determine if the test statistic falls into the rejection region

or not. Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests1. Determine the parameter of interest.

2. Determine the null hypothesis H0.3. Determine the alternative hypothesis Ha.4. Choose the appropriate test statistic.5. Determine the rejection region using the significance level.6. Determine if the test statistic falls into the rejection region

or not. Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests1. Determine the parameter of interest.2. Determine the null hypothesis H0.

3. Determine the alternative hypothesis Ha.4. Choose the appropriate test statistic.5. Determine the rejection region using the significance level.6. Determine if the test statistic falls into the rejection region

or not. Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests1. Determine the parameter of interest.2. Determine the null hypothesis H0.3. Determine the alternative hypothesis Ha.

4. Choose the appropriate test statistic.5. Determine the rejection region using the significance level.6. Determine if the test statistic falls into the rejection region

or not. Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests1. Determine the parameter of interest.2. Determine the null hypothesis H0.3. Determine the alternative hypothesis Ha.4. Choose the appropriate test statistic.

5. Determine the rejection region using the significance level.6. Determine if the test statistic falls into the rejection region

or not. Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests1. Determine the parameter of interest.2. Determine the null hypothesis H0.3. Determine the alternative hypothesis Ha.4. Choose the appropriate test statistic.5. Determine the rejection region using the significance level.

6. Determine if the test statistic falls into the rejection regionor not. Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests1. Determine the parameter of interest.2. Determine the null hypothesis H0.3. Determine the alternative hypothesis Ha.4. Choose the appropriate test statistic.5. Determine the rejection region using the significance level.6. Determine if the test statistic falls into the rejection region

or not.

Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests1. Determine the parameter of interest.2. Determine the null hypothesis H0.3. Determine the alternative hypothesis Ha.4. Choose the appropriate test statistic.5. Determine the rejection region using the significance level.6. Determine if the test statistic falls into the rejection region

or not. Reject or don’t reject accordingly.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example.

A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet.

The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact.

The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in.

Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in.

Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest:

µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.

Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis:

H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)

Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis:

Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in

(the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic:

z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n

=x−1.9

0.06/√

n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. A new type of body armor is tested if it satisfies thespecification of at most µ0 = 1.9in of displacement when hitwith a certain type of bullet. The manufacturer tests by firingone round each at 36 samples of the new armor and measuringthe displacement upon impact. The result is a sample meandisplacement of 1.91 in. Assume the displacements arenormally distributed with mean µ and a standard deviation of0.06 in. Test if the armor is up to specifications at the 10%significance level.

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region:

We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.

Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:

z =x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z

=x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n

=1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not.

Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ , the true mean displacement.Null hypothesis: H0 : µ = 1.9in (null value of µ0)Alternative hypothesis: Ha : µ > 1.9in (the main concern is adisplacement that is larger than what is claimed).

Test statistic: z =x−µ0

σ/√

n=

x−1.90.06/

√n

Rejection region: We use an upper tailed test and the rejectionregion is z > z0.1 ≈ 1.28.Substitute values into test statistic:z =

x−µ0

σ/√

n=

1.91−1.90.06/

√36

=0.010.01

= 1

Decide if to reject or not. Do not reject the null hypothesis,because the test statistic is not in the rejection region.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

A type II error is more serious than a type I error here. At thesame time, it is sensible to use a null hypothesis that says yourproduct works. Otherwise false negatives may keep you fromever producing.

But when a test says “reject”, you really reject. That’s it. Evenif it’s expensive.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

DiscussionA type II error is more serious than a type I error here.

At thesame time, it is sensible to use a null hypothesis that says yourproduct works. Otherwise false negatives may keep you fromever producing.

But when a test says “reject”, you really reject. That’s it. Evenif it’s expensive.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

DiscussionA type II error is more serious than a type I error here. At thesame time, it is sensible to use a null hypothesis that says yourproduct works.

Otherwise false negatives may keep you fromever producing.

But when a test says “reject”, you really reject. That’s it. Evenif it’s expensive.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

DiscussionA type II error is more serious than a type I error here. At thesame time, it is sensible to use a null hypothesis that says yourproduct works. Otherwise false negatives may keep you fromever producing.

But when a test says “reject”, you really reject. That’s it. Evenif it’s expensive.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

DiscussionA type II error is more serious than a type I error here. At thesame time, it is sensible to use a null hypothesis that says yourproduct works. Otherwise false negatives may keep you fromever producing.

But when a test says “reject”, you really reject.

That’s it. Evenif it’s expensive.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

DiscussionA type II error is more serious than a type I error here. At thesame time, it is sensible to use a null hypothesis that says yourproduct works. Otherwise false negatives may keep you fromever producing.

But when a test says “reject”, you really reject. That’s it.

Evenif it’s expensive.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

DiscussionA type II error is more serious than a type I error here. At thesame time, it is sensible to use a null hypothesis that says yourproduct works. Otherwise false negatives may keep you fromever producing.

But when a test says “reject”, you really reject. That’s it. Evenif it’s expensive.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′)

= P(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)

= P(

X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)

= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:

If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β (µ′) = P

(X < µ0 + zασ/

√n|µ = µ

′)= P

(X−µ ′

σ/√

n< zα +

µ0−µ ′

σ/√

n

)= Φ

(zα +

µ0−µ ′

σ/√

n

)

-

µ0

α

µ ’

β

Upper tail test for µ≤µ0:If µ=µ ’>µ0, non-rejection probability is β .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities,

provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given.

For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size thatwill yield these probabilities, provided that a value µ ′ for whichβ (µ ′) = β is given. For an upper tailed test, we should choosen to satisfy the following.

Φ

(zα +

µ0−µ ′

σ/√

n

)= β

zα +µ0−µ ′

σ/√

n= −zβ

zα + zβ = −√

nµ0−µ ′

σ

√n =

σ(zα + zβ )µ ′−µ0

n =(

σ(zα + zβ )µ ′−µ0

)2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example.

Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet.

Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in

(military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in)

, normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in

a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28

z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n

=

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉

= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e

= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. Suppose a production run of a new type of bodyarmor is tested if it satisfies the specification of at mostµ0 = 1.9 in of displacement when hit with a certain type ofbullet. Compute n so that with µ ′ = 2 in (military spec:displacement no more than 2 in), normal distribution andσ = 0.06 in a test at the 10% significance level has at most achance of 0.0005 = 0.05% for a type II error.

z0.1 = 1.28z0.0005 = 3.29

n =

⌈(0.06(1.28+3.29)

2−1.9

)2⌉

=⌈(0.6 ·4.57)2

⌉= d7.52e= 8

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

1. Usually, σ is not known.2. For large sample size, use the same tests, but use the

sample statistic z =x−µ0

s/√

n.

This can be done because, by the Central Limit Theorem,

for large enough n, the random variable Z =X−µ

S/√

nis

approximately normally distributed.Rule of thumb: n > 40.Computations and formulas are the same. For β andsample size, we would need to assume we know σ .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size1. Usually, σ is not known.

2. For large sample size, use the same tests, but use the

sample statistic z =x−µ0

s/√

n.

This can be done because, by the Central Limit Theorem,

for large enough n, the random variable Z =X−µ

S/√

nis

approximately normally distributed.Rule of thumb: n > 40.Computations and formulas are the same. For β andsample size, we would need to assume we know σ .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size1. Usually, σ is not known.2. For large sample size, use the same tests, but use the

sample statistic z =x−µ0

s/√

n.

This can be done because, by the Central Limit Theorem,

for large enough n, the random variable Z =X−µ

S/√

nis

approximately normally distributed.Rule of thumb: n > 40.Computations and formulas are the same. For β andsample size, we would need to assume we know σ .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size1. Usually, σ is not known.2. For large sample size, use the same tests, but use the

sample statistic z =x−µ0

s/√

n.

This can be done because, by the Central Limit Theorem,

for large enough n, the random variable Z =X−µ

S/√

nis

approximately normally distributed.

Rule of thumb: n > 40.Computations and formulas are the same. For β andsample size, we would need to assume we know σ .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size1. Usually, σ is not known.2. For large sample size, use the same tests, but use the

sample statistic z =x−µ0

s/√

n.

This can be done because, by the Central Limit Theorem,

for large enough n, the random variable Z =X−µ

S/√

nis

approximately normally distributed.Rule of thumb: n > 40.

Computations and formulas are the same. For β andsample size, we would need to assume we know σ .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size1. Usually, σ is not known.2. For large sample size, use the same tests, but use the

sample statistic z =x−µ0

s/√

n.

This can be done because, by the Central Limit Theorem,

for large enough n, the random variable Z =X−µ

S/√

nis

approximately normally distributed.Rule of thumb: n > 40.Computations and formulas are the same.

For β andsample size, we would need to assume we know σ .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size1. Usually, σ is not known.2. For large sample size, use the same tests, but use the

sample statistic z =x−µ0

s/√

n.

This can be done because, by the Central Limit Theorem,

for large enough n, the random variable Z =X−µ

S/√

nis

approximately normally distributed.Rule of thumb: n > 40.Computations and formulas are the same. For β andsample size, we would need to assume we know σ .

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

3. For small samples from a normal distribution, use theone-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.

Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

n

Alternative hypotheses and rejection regions:I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)

I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)

I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)

Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size3. For small samples from a normal distribution, use the

one-sample t-test.Null hypothesis: H0 : µ = µ0

Test statistic: t =x−µ0

s/√

nAlternative hypotheses and rejection regions:

I Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test)I Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test)I Ha : µ 6= µ0, t ≤−t α

2 ,n−1 or t ≥ t α

2 ,n−1 (two tailed test)Alternative hypotheses and rejection regions are often tabulatedin statistics texts for more convenient reference.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

1. Computing β (µ ′) for t-tests is complicated, because µ and

σ are both unknown. To get β (µ ′), compute d =|µ0−µ ′|

σwith some estimate for σ . Then use the β curves (in yourstatistics book) with the right number of degrees offreedom to get β .

2. To choose the sample size, find d and your desired β .Estimate which curve (determined by the degrees offreedom) should be at (d,β ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests1. Computing β (µ ′) for t-tests is complicated

, because µ and

σ are both unknown. To get β (µ ′), compute d =|µ0−µ ′|

σwith some estimate for σ . Then use the β curves (in yourstatistics book) with the right number of degrees offreedom to get β .

2. To choose the sample size, find d and your desired β .Estimate which curve (determined by the degrees offreedom) should be at (d,β ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests1. Computing β (µ ′) for t-tests is complicated, because µ and

σ are both unknown.

To get β (µ ′), compute d =|µ0−µ ′|

σwith some estimate for σ . Then use the β curves (in yourstatistics book) with the right number of degrees offreedom to get β .

2. To choose the sample size, find d and your desired β .Estimate which curve (determined by the degrees offreedom) should be at (d,β ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests1. Computing β (µ ′) for t-tests is complicated, because µ and

σ are both unknown. To get β (µ ′), compute d =|µ0−µ ′|

σwith some estimate for σ .

Then use the β curves (in yourstatistics book) with the right number of degrees offreedom to get β .

2. To choose the sample size, find d and your desired β .Estimate which curve (determined by the degrees offreedom) should be at (d,β ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests1. Computing β (µ ′) for t-tests is complicated, because µ and

σ are both unknown. To get β (µ ′), compute d =|µ0−µ ′|

σwith some estimate for σ . Then use the β curves (in yourstatistics book) with the right number of degrees offreedom to get β .

2. To choose the sample size, find d and your desired β .Estimate which curve (determined by the degrees offreedom) should be at (d,β ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests1. Computing β (µ ′) for t-tests is complicated, because µ and

σ are both unknown. To get β (µ ′), compute d =|µ0−µ ′|

σwith some estimate for σ . Then use the β curves (in yourstatistics book) with the right number of degrees offreedom to get β .

2. To choose the sample size, find d and your desired β .

Estimate which curve (determined by the degrees offreedom) should be at (d,β ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests1. Computing β (µ ′) for t-tests is complicated, because µ and

σ are both unknown. To get β (µ ′), compute d =|µ0−µ ′|

σwith some estimate for σ . Then use the β curves (in yourstatistics book) with the right number of degrees offreedom to get β .

2. To choose the sample size, find d and your desired β .Estimate which curve (determined by the degrees offreedom) should be at (d,β ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example.

(DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a)

A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon.

The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows

:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4.

Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100

Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

n

Rejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201

Values: x = 98.375, s≈ 6.109, s/√

n≈ 1.764, T =−0.921Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values

: x = 98.375, s≈ 6.109, s/√

n≈ 1.764, T =−0.921Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375

, s≈ 6.109, s/√

n≈ 1.764, T =−0.921Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109

, s/√

n≈ 1.764, T =−0.921Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764

, T =−0.921Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. The resulting readings were as follows:105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7,103.3, 92.4. Does this data suggest that the population meanreading under these conditions differs from 100? State and testthe appropriate hypothesis at the 0.05 significance level.

Parameter: mean reading µ

H0 : µ = 100Ha : µ 6= 100 (Could use µ < 100. Definitely not µ > 100.)

Test statistic: T =x−µ0

s/√

nRejection region: For α = 0.05, |T|> t0.025,11 ≈ 2.201Values: x = 98.375, s≈ 6.109, s/

√n≈ 1.764, T =−0.921

Decision: Don’t reject.Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon.

Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed. How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ=

23

, α = 0.05

n−1≈ 39 (tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed.

How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ=

23

, α = 0.05

n−1≈ 39 (tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed. How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ=

23

, α = 0.05

n−1≈ 39 (tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed. How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ

=23

, α = 0.05

n−1≈ 39 (tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed. How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ=

23

, α = 0.05

n−1≈ 39 (tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed. How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ=

23

, α = 0.05

n−1≈ 39 (tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed. How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ=

23

, α = 0.05

n−1≈ 39

(tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means

logo1

Testing Normal Distributions Example Sample Size Determination t-tests

Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radondetectors of a certain type was selected, and each was exposedto 100 pCi/L of radon. Suppose that prior to the experiment, avalue of σ = 7.5 pCi/L had been assumed. How manydeterminations would then have been appropriate to obtainβ = 0.1 for the alternative µ = 95?

d =|µ ′−µ0|

σ=

23

, α = 0.05

n−1≈ 39 (tables are wonderful)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Hypothesis Tests for Population Means