Lecture 11 Dustin 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 H 1 : μ≠100, p-value>.05 2.When testing H 1 : μ≠100, p-value<.05 STA 291 Winter 09/10 Lecture 112

 If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 1% level of significance? 1.Yes 2.No 3.Maybe STA 291 Winter 09/10 Lecture 113

 If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 5% level of significance? 1.Yes 2.No 3.Maybe STA 291 Winter 09/10 Lecture 114

 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 H 0 : μ=μ 0 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 5STA 291 Winter 09/10 Lecture 11

 Same process as with population mean  Value we are testing against is called p 0  Test statistic  P-value ◦ Calculation is exactly the same as for the test for a mean  Sample size restrictions: 6STA 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 H 0 : 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 7STA 291 Winter 09/10 Lecture 11

 Similar to testing one proportion  Hypotheses are set up like two sample mean test ◦ H 0 :p 1 =p 2  Same as H 0 :p 1 -p 2 =0  Test Statistic 8STA 291 Winter 09/10 Lecture 11

 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. STA 291 Winter 09/10 Lecture 119

 Similar to testing one proportion  Hypotheses are set up like two sample mean test ◦ H 0 :p 1 -p 2 =0  Same as H 0 : p 1 =p 2  Test Statistic 10STA 291 Winter 09/10 Lecture 11

 Hypothesis involves 2 parameters from 2 populations ◦ Test statistic is different  Involves 2 large samples (both samples at least 30)  One from each population  H 0 : μ 1 -μ 2 =0 ◦ Same as H 0 : μ 1 =μ 2 ◦ Test statistic 11STA 291 Winter 09/10 Lecture 11

 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. 12STA 291 Winter 09/10 Lecture 11

 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 13STA 291 Winter 09/10 Lecture 11

 Test Statistic ◦ Degrees of freedom  n 1 +n STA 291 Winter 09/10 Lecture 11

◦ Degrees of freedom  n 1 +n STA 291 Winter 09/10 Lecture 11

 Test statistic  Degrees of freedom 16STA 291 Winter 09/10 Lecture 11

17STA 291 Winter 09/10 Lecture 11

18  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