STA 291Winter 09/10

Lecture 11Dustin Lueker

A 95% confidence interval for µ is (96,110). Which of the following statements about significance tests for the same data is true?

1. When testing H1: μ≠100, p-value>.052. When testing H1: μ≠100, p-value<.05

Example

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If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 1% level of significance?1. Yes2. No3. Maybe

Example

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If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 5% level of significance?1. Yes2. No3. Maybe

Example

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Results of confidence intervals and of two-sided significance tests are consistent◦ Whenever the hypothesized mean is not in the

confidence interval around the sample mean, then the p-value for testing H0: μ=μ0 is smaller than 5% (significance at the 5% level)

◦ In general, a 100(1-α)% confidence interval corresponds to a test at significance level α

Correlation Between Tests and Confidence Intervals

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Same process as with population mean Value we are testing against is called p0 Test statistic

P-value◦ Calculation is exactly the same as for the test for

a mean Sample size restrictions:

Significance Test for a Proportion

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STA 291 Winter 09/10 Lecture 11

Let p denote the proportion of Floridians who think that government environmental regulations are too strict

A telephone poll of 824 people conducted in June 1995 revealed that 26.6% said regulations were too strict◦ Test H0: p=.5 at α=.05◦ Calculate the test statistic◦ Find the p-value and interpret

Construct a 95% confidence interval. What is the advantage of the confidence interval over the test

Example

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Testing Difference Between Two Population Proportions Similar to testing one proportion Hypotheses are set up like two sample

mean test◦ H0:p1=p2

Same as H0:p1-p2=0 Test Statistic

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Government agencies have undertaken surveys of Americans 12 years of age and older. Each was asked whether he or she used drugs at least once in the past month. The results of this year’s survey had 171 yes responses out of 306 surveyed while the survey 10 years ago resulted in 158 yes responses out of 304 surveyed. Test whether the use of drugs in the past ten years has increased.

State and test the hypotheses using the rejection region method at the 5% level of significance.

Example

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Testing Difference Between Two Population Proportions Similar to testing one proportion Hypotheses are set up like two sample

mean test◦ H0:p1-p2=0

Same as H0: p1=p2

Test Statistic

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Testing the Difference Between Means from Different Populations Hypothesis involves 2 parameters from 2

populations◦ Test statistic is different

Involves 2 large samples (both samples at least 30) One from each population

H0: μ1-μ2=0◦ Same as H0: μ1=μ2◦ Test statistic

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Example In the 1982 General Social Survey, 350

subjects reported the time spend every day watching television. The sample yielded a mean of 4.1 and a standard deviation of 3.3.

In the 1994 survey, 1965 subjects yielded a sample mean of 2.8 hours with a standard deviation of 2.◦ Set up hypotheses of a significance test to

analyze whether the population means differ in 1982 and 1994 and test at α=.05 using the p-value method.

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Small Sample Tests for Two Means Used when comparing means of two

samples where at least one of them is less than 30◦ Normal population distribution is assumed for

both samples Equal Variances

◦ Both groups have the same variability

Unequal Variances◦ Both groups may not have the same variability

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Small Sample Test for Two Means, Equal Variances Test Statistic

◦ Degrees of freedom n1+n2-2

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Small Sample Confidence Interval for Two Means, Equal Variances

◦ Degrees of freedom n1+n2-2

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Small Sample Test for Two Means, Unequal Variances Test statistic

Degrees of freedom

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Small Sample Confidence Interval for Two Means, Unequal Variances

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Method 1 (Equal Variances) vs. Method 2 (Unequal Variances)

How to choose between Method 1 and Method 2?◦ Method 2 is always safer to use◦ Definitely use Method 2

If one standard deviation is at least twice the other If the standard deviation is larger for the sample with

the smaller sample size◦ Usually, both methods yield similar conclusions

STA 291 Winter 09/10 Lecture 11