1 Confidence Interval and Hypothesis Testing for: Population Mean ( )

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Confidence Interval and Hypothesis Testing for:

Population Mean ( )

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Assumptions & ConditionsRandom sample

Independent observations

Nearly normal distribution

y ~ N (, / n )

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y - sn

t =

Student’s t-Model for decisions about the mean,

With df=n-1

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One-Sample t-IntervalWhen the conditions are met, the confidence interval for the means of one population is:

_______________________

where the standard error of the means is:_______________________

The critical value (t*) depends on the particular confidence level, C, and the degrees of freedom, df.

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CI for the mean,

Stat, tests, 8: T-Interval

HT for the mean,

Stat, tests, 2: T-Test

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Terms

Significant Level ()

P-value (P in TI)

Null Hypothesis (H

Alternative Hypothesis (H

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Reject the null hypothesis if the P-value is less than or equal to the significance level .

Reject Ho if P-value < .

Fail to reject Ho if P-value > .

Decisions

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Procedures1. Hypotheses:

H

H

2. Assumptions and Conditions:

3. Mechanics

T = P-value =< Significant Level ()?

4. Conclusion: Answer the original question.

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Confidence Interval and Hypothesis

TestingComparing Two Population

Means: Finding and Testing their

difference (1- 2)

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Assumptions and Conditions for t-model• Independence Assumption (Each condition needs to be checked for both groups.):– Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)?

– 10% Condition: Is the sample size (n) less than 10% of the population size (N)? We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample.

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Assumptions and Conditions (cont.)

• Normal Population Assumption:– Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition.

• Independent Groups Assumption: The two groups we are comparing must be independent of each other.

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Two-Sample t-IntervalWhen the conditions are met, the confidence interval for the difference (between means of two independent groups) is:

where the standard error of the difference of the means is:

The critical value (t*) depends on the particular

confidence level, C, and the degrees of freedom, df, derived from the sample sizes and a special formula.

2 21 2

1 21 2

s sSE y y

n n

1 2 1 2dfy y t SE y y

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Degrees of Freedom (df)

• The special formula for the degrees of freedom for our t critical value is a bear:

• Because of this, we will let technology calculate degrees of freedom for us!

(or pursue a stat major or minor)

22 21 2

1 22 22 2

1 2

1 1 2 2

1 11 1

s sn n

dfs s

n n n n

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Testing the Difference Between Two Means

• The hypothesis test we use is the two-sample t-test for means.• The conditions for the two-sample t-test for the difference between the means of two independent groups are the same as for the two-sample t-interval.

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Testing the Difference Between Two Means

(cont.)We test the hypothesis H0:1 – 2 = 0, where the hypothesized difference, 0, is almost always 0, using the statistic

The standard error is

When the conditions are met and the null hypothesis is true, this statistic can be closely modeled by a Student’s t-model with a number of degrees of freedom given by a special formula. We use that model to obtain a P-value.

1 2 0

1 2

y yt

SE y y

2 21 2

1 21 2

s sSE y y

n n

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Back Into the Pool• Remember that when we know a proportion, we know its standard deviation. – Thus, when testing the null hypothesis that two proportions were equal, we could assume their variances were equal as well.

– This led us to pool our data for the hypothesis test for p1-p2.

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Back Into the Pool (cont.)

• For means, there is also a pooled t-test.– Like the two-proportions z-test, this test assumes that the variances in the two groups are equal.

– But, be careful, there is no link between a mean and its standard deviation…

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Back Into the Pool (cont.)

• If we are willing to assume that the variances of two means are equal, we can pool the data from two groups to estimate the common variance and make the degrees of freedom formula much simpler.

• We are still estimating the pooled standard deviation from the data, so we use Student’s t-model, and the test is called a pooled t-test.

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*The Pooled t-Test• If we assume that the variances are equal, we can estimate the common variance from the numbers we already have:

• Substituting into our standard error formula, we get:

• Our degrees of freedom are now df = n1 + n2 – 2.

2 21 1 2 22

1 2

1 1

1 1pooled

n s n ss

n n

1 21 2

1 1pooled pooledSE y y s

n n

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*The Pooled t-Test and Confidence Interval

• The conditions are the same, plus the assumption that the variances of the two groups are the same.

• For the hypothesis test, our test statistic is

which has df = n1 + n2 – 2.

• Our confidence interval is

1 2 0

1 2pooled

y yt

SE y y

1 2 1 2df pooledy y t SE y y

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Is the Pool All Wet?• So, when should you use pooled-t methods rather than two-sample t methods? Well, hardly ever.

• Because the advantages of pooling are small, and you are allowed to pool only rarely (when the equal variance assumption is met).

• Don’t pool.

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Can We Test Whether the Variances Are

Equal?

• The test is very sensitive to non-normal data and works poorly for small sample sizes.

• So, the test does not work when we need it to.

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What Can Go Wrong?• Watch out for paired data.

– The Independent Groups Assumption deserves special attention.

– If the samples are not independent, you can’t use two-sample methods.

• Look at the plots.– Check for outliers and non-normal distributions by making and examining boxplots or normal probability plots.

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What have we learned? To use statistical inference to compare the means of two independent groups.– We use t-models for CI and HT.– It is important to check conditions to see if the assumptions for t-model are met.

– Don’t pool the standard errors.

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Coke Versus Pepsi

Independent random samples of 36 cans of Coke and Pepsi are weighed and summarized below. Use the 0.01 significance level to test the claim that the mean weight of regular Coke is different from the mean weight of regular Pepsi.

Regular Coke Regular Pepsi

n 36 36

y 0.817 0.824

s 0.0076 0.0057

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Coke Versus Pepsi

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Ho : 1 = 2Ha : 1 2 = 0.01

Coke vs. Pepsi

Fail to reject H0Reject H0Reject H0

- t* = - ____ t* = _____ t = 0