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7 Elementary Statistics Larson Farber Hypothesis Testing
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Page 1: 7 Elementary Statistics Larson Farber Hypothesis Testing.

7

Elementary Statistics

Larson Farber

Hypothesis Testing

Page 2: 7 Elementary Statistics Larson Farber Hypothesis Testing.

Introduction to Hypothesis

Testing

Section 7.1

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A statistical hypothesis is a claim about a population.

Alternative hypothesis Ha

contains a statement of inequality such as < , or >

Null hypothesis H0

contains a statement of equality such as , = or

.

Complementary Statements

If I am false, you are true

If I am false, you are true

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A consumer magazine claims the proportion of cell phone calls made during evenings and weekends is at most 60%.

Write the claim about the population. Then, write its complement. Either hypothesis, the null or the alternative, can represent the claim.

A hospital claims its ambulance response time is less than 10 minutes.

Writing Hypotheses

claim

claim

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Begin by assuming the equality condition in the null hypothesis is true. This is regardless of whether the claim is represented by the null hypothesis or by the alternative hypothesis.

Hypothesis Test Strategy

Collect data from a random sample taken from the population and calculate the necessary sample statistics.

If the sample statistic has a low probability of being drawn from a population in which the null hypothesis is true, you willreject H0. (As a consequence, you will support the alternativehypothesis.) If the probability is not low enough, fail to reject H0.

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A type I error: Null hypothesis is actually true but the decision is to reject it.

Level of significance, Maximum probability of committing a type I error.

Actual Truth of H0

Errors and Level of Significance

H0 True H0 FalseDo notreject H0

Reject H0

CorrectDecision

CorrectDecision

Type IIError

Type IErrorD

ecis

ion

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Right-tail test

Two-tail test

Left-tail test

Types of Hypothesis Tests

Ha is more probable

Ha is more probable

Ha is more probable

Page 8: 7 Elementary Statistics Larson Farber Hypothesis Testing.

The P-value is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined by the sample data.

If z is negative, twice the area in the left tail

If z is positive, twice the area in the right tail

P-values

P-value = indicated area

z z

zz

Area inleft tail

Area inright tail

For a left tail test For a right tail test

For a two-tail test

Page 9: 7 Elementary Statistics Larson Farber Hypothesis Testing.

Finding P-values: 1-tail Test

The test statistic for a right-tail test is z = 1.56. Find the P-value.

The area to the right of z = 1.56 is 1 – .9406 = 0.0594.The P-value is 0.0594.

z = 1.56

Area in right tail

Page 10: 7 Elementary Statistics Larson Farber Hypothesis Testing.

The test statistic for a two-tail test is z = –2.63. Find the corresponding P-value.

The area to the left of z = –2.63 is 0.0043.The P-value is 2(0.0043) = 0.0086.

Finding P-values: 2-tail Test

z = –2.63

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Test Decisions with P-values

The decision about whether there is enough evidence to reject the null hypothesis can be made by comparing the P-value to the value of ,the level of significance of the test.

If fail to reject the null hypothesis.

If reject the null hypothesis.

Page 12: 7 Elementary Statistics Larson Farber Hypothesis Testing.

The P-value of a hypothesis test is 0.0749. Make your decision at the 0.05 level of significance.

Compare the P-value to . Since 0.0749 > 0.05, fail to reject H0.

If P = 0.0246, what is your decision if

1) Since , reject H0.

2) Since 0.0246 > 0.01, fail to reject H0.

Using P-values

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There is enough evidence to

reject the claim.

Claim

Interpreting the Decision

Claim is H0 Claim is Ha

Reject H0

Fail to reject H0

Dec

isio

n

There is not enough

evidence to reject the claim.

There is enough evidence to support the

claim.

There is not enough

evidence to support the

claim.

Page 14: 7 Elementary Statistics Larson Farber Hypothesis Testing.

1. Write the null and alternative hypothesis.

2. State the level of significance.

3. Identify the sampling distribution.

Write H0 and Ha as mathematical statements. Remember H0 always contains the = symbol.

This is the maximum probability of rejecting the null hypothesis when it is actually true. (Making a type I error.)

The sampling distribution is the distribution for the test statistic assuming that the equality condition in H0 is true and that the experiment is repeated an infinite number of times.

Steps in a Hypothesis Test

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4. Find the test statistic and standardize it.

Perform the calculations to standardize your sample statistic.

5. Calculate the P-value for the test statistic.

This is the probability of obtaining your test statistic or one that is more extreme from the sampling distribution.

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If the P-value is less than (the level of significance) reject H0. If the P value is greater , fail to reject H0.

6. Make your decision.

7. Interpret your decision.

If the claim is the null hypothesis, you will either reject the claim or determine there is not enough evidence to reject the claim.

If the claim is the alternative hypothesis, you will either support the claim or determine there is not enough evidence to support the claim.

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Hypothesis Testing for the

Mean (n 30)

Section 7.2

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The z-Test for a MeanThe z-test is a statistical test for a population mean. The z-test can be used: (1) if the population is normal and s is known or (2) when the sample size, n, is at least 30. The test statistic is the sample mean and the standardizedtest statistic is z.

When n 30, use s in place of .

Page 19: 7 Elementary Statistics Larson Farber Hypothesis Testing.

A cereal company claims the mean sodium content in one serving of its cereal is no more than 230 mg. You work for a national health service and are asked to test this claim. You find that a random sample of 52 servings has a mean sodium content of 232 mg and a standard deviation of 10 mg. At = 0.05, do you have enough evidence to reject the company’s claim?

1. Write the null and alternative hypothesis.

2. State the level of significance. = 0.05

3. Determine the sampling distribution.

Since the sample size is at least 30, the sampling distribution is normal.

The z-Test for a Mean (P-value)

Page 20: 7 Elementary Statistics Larson Farber Hypothesis Testing.

4. Find the test statistic and standardize it.

5. Calculate the P-value for the test statistic.

Since this is a right-tail test, the P-value is the area found to the rightof z = 1.44 in the normal distribution. From the table P = 1 – 0.9251

n = 52

s = 10

Test statistic

z = 1.44

Area in right tail

P = 0.0749.

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6. Make your decision.

7. Interpret your decision.

Compare the P-value to . Since 0.0749 > 0.05, fail to reject H0.

There is not enough evidence to reject the claim that the mean sodium content of one serving of its cereal is no more than 230 mg.

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Sampling distribution for

The rejection region is the range of values for which the null hypothesis is not probable. It is always in the direction of the alternative hypothesis. Its area is equal to . A critical value separates the rejection region from the non-rejection region.

Rejection Regions

Rejection Region

Critical Value z0z z0

Page 23: 7 Elementary Statistics Larson Farber Hypothesis Testing.

The critical value z0 separates the rejection region from the non-rejection region. The area of the rejection region is .

Find z0 for a left-tail

test with = .01.

Find z0 for a right-tail

test with = .05.

Find –z0 and z0 for a two-tail test with = .01.

z0 = –2.33–z0 = –2.575 and z0 = 2.575

z0 = 1.645

Critical Values

z0 z0

Rejectionregion

Rejectionregion

z0z0

Rejectionregion

Rejectionregion

Page 24: 7 Elementary Statistics Larson Farber Hypothesis Testing.

1. Write the null and alternative hypothesis.

2. State the level of significance.

3. Identify the sampling distribution.

Write H0 and Ha as mathematical statements. Remember H0 always contains the = symbol.

This is the maximum probability of rejecting the null hypothesis when it is actually true. (Making a type I error.)

The sampling distribution is the distribution for the test statistic assuming that the equality condition in H0 is true and that the experiment is repeated an infinite number of times.

Using the Critical Value to Make Test Decisions

Page 25: 7 Elementary Statistics Larson Farber Hypothesis Testing.

6. Find the test statistic.

5. Find the rejection region.

4. Find the critical value.

The critical value separates the rejection region of the sampling distribution from the non-rejection region. The area of the critical region is equal to the level of significance of the test.

Perform the calculations to standardize your sample statistic.

z0

Rejection Region

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7. Make your decision.

8. Interpret your decision.

If the test statistic falls in the critical region, reject H0. Otherwise, fail to reject H0.

If the claim is the null hypothesis, you will either reject the claim or determine there is not enough evidence to reject the claim.

If the claim is the alternative hypothesis, you will either support the claim or determine there is not enough evidence to support the claim.

Page 27: 7 Elementary Statistics Larson Farber Hypothesis Testing.

A cereal company claims the mean sodium content in one serving of its cereal is no more than 230 mg. You work for a national health service and are asked to test this claim. You find that a random sample of 52 servings has a mean sodium content of 232 mg and a standard deviation of 10 mg. At = 0.05, do you have enough evidence to reject the company’s claim?

2. State the level of significance. = 0.05

3. Determine the sampling distribution.

Since the sample size is at least 30, the sampling distribution is normal.

The z-Test for a Mean

1. Write the null and alternative hypothesis.

Page 28: 7 Elementary Statistics Larson Farber Hypothesis Testing.

n = 52 = 232 s = 10

7. Make your decision.

6. Find the test statistic and standardize it.

8. Interpret your decision.

5. Find the rejection region.

Rejectionregion

Since Ha contains the > symbol, this is a right-tail test.

z = 1.44 does not fall in the rejection region, so fail to reject H0

There is not enough evidence to reject the company’s claim that there is at most 230 mg of sodium in one serving of its cereal.

1.645

4. Find the critical value.

z0

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Using the P-value of a Test to Compare Areas

z0

Rejection area0.05

z0 = –1.645

z

Area to th

e left o

f z

0.1093

z = –1.23

For a critical value decision, decide if z is in the rejection region If z is in the rejection region, reject H0. If z is not in the rejection

region, fail to reject H0.

= 0.05

For a P-value decision, compare areas.

If reject H0. If fail to reject H0.

P = 0.1093

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Hypothesis Testing for the

Mean (n < 30)

Section 7.3

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Find the critical value t0 for a left-tailed test given = 0.01 and n = 18.

Find the critical values –t0 and t0 for a two-tailed test given

d.f. = 18 – 1 = 17

t0

t0 = –2.567

d.f. = 11 – 1 = 10

–t0 = –2.228 and t0 = 2.228

The t Sampling Distribution

= 0.05 and n = 11.

Area inleft tail

t0 t0

Page 32: 7 Elementary Statistics Larson Farber Hypothesis Testing.

A university says the mean number of classroom hours per week for full-time faculty is 11.0. A random sample of the number of classroom hours for full-time faculty for one week is listed below. You work for a student organization and are asked to test this claim. At = 0.01, do you have enough evidence to reject the university’s claim?11.8 8.6 12.6 7.9 6.4 10.4 13.6 9.1

1. Write the null and alternative hypothesis

2. State the level of significance = 0.01

3. Determine the sampling distribution

Since the sample size is 8, the sampling distribution is a t-distribution with 8 – 1 = 7 d.f.

Testing –Small Sample

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t = –1.08 does not fall in the rejection region, so fail to reject H0 at = 0.01

n = 8 = 10.050 s = 2.485

7. Make your decision.

6. Find the test statistic and standardize it

8. Interpret your decision.

There is not enough evidence to reject the university’s claim that faculty spend a mean of 11 classroom hours.

5. Find the rejection region.

Since Ha contains the ≠ symbol, this is a two-tail test.

4. Find the critical values.

–3.499 3.499t0–t0

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T-Test of the Mean

Test of = 11.000 vs not = 11.000

Variable N Mean StDev SE Mean T PHours 8 0.050 2.485 0.879 –1.08 0.32

Enter the data in C1, ‘Hours’. Choose t-test in the STAT menu.

Minitab reports the t-statistic and the P-value.

Since the P-value is greater than the level of significance (0.32 > 0.01), fail to reject the null hypothesis at the 0.01 level of significance.

Minitab Solution

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Hypothesis Testing for Proportions

Section 7.4

Page 36: 7 Elementary Statistics Larson Farber Hypothesis Testing.

p is the population proportion of successes. The

test statistic is .

If and the sampling distribution for is normal.

Test for Proportions

The standardized test statistic is:

(the proportion of sample successes)

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Test for ProportionsA communications industry spokesperson claims that over 40% of Americans either own a cellular phone or have a family member who does. In a random survey of 1036 Americans, 456 said they or a family member owned a cellular phone. Test the spokesperson’s claim at = 0.05. What can you conclude?

1. Write the null and alternative hypothesis.

2. State the level of significance.

= 0.05

Page 38: 7 Elementary Statistics Larson Farber Hypothesis Testing.

3. Determine the sampling distribution.

7. Make your decision.

6. Find the test statistic and standardize it.

8. Interpret your decision.

z = 2.63 falls in the rejection region, so reject H0

There is enough evidence to support the claim that over 40% of Americans own a cell phone or have a family member who does.

1036(.40) > 5 and 1036(.60) > 5. The sampling distribution is normal.

n = 1036 x = 456

4. Find the critical value.

1.645

5. Find the rejection region.

Rejectionregion

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Hypothesis Testing for

Variance and Standard Deviation

Section 7.5

Page 40: 7 Elementary Statistics Larson Farber Hypothesis Testing.

Find a 20 critical value for a left-tail test when n = 17 and = 0.05.

s2 is the test statistic for the population variance. Its sampling distribution is a 2 distribution with n – 1 d.f.

Find critical values 20 for a two-tailed test when n = 12, = 0.01.

The standardized test statistic is

20 = 7.962

2L = 2.603 and 2

R = 26.757

Critical Values for

Page 41: 7 Elementary Statistics Larson Farber Hypothesis Testing.

A state school administrator says that the standard deviation of test scores for 8th grade students who took a life-science assessment test is less than 30. You work for the administrator and are asked to test this claim. You find that a random sample of 10 tests has a standard deviation of 28.8. At = 0.01, do you have enough evidence to support the administrator’s claim? Assume test scores are normally distributed.

1. Write the null and alternative hypothesis.

2. State the level of significance.

= 0.013. Determine the sampling distribution.

The sampling distribution is 2 with 10 – 1 = 9 d.f.

Test for

Page 42: 7 Elementary Statistics Larson Farber Hypothesis Testing.

7. Make your decision.

6. Find the test statistic.

8. Interpret your decision.

n = 10s = 28.8

2 = 8.2944 does not fall in the rejection region, so fail to reject

H0

There is not enough evidence to support the administrator’s claim that the standard deviation is less than 30.

2.088

4. Find the critical value.

5. Find the rejection region.