STA 291 Summer 2010

1. STA 291Summer 2010 Lecture 18 Dustin Lueker

2. Example • The p-value for testing H1: µ≠100 is p=.001. This indicates that… • There is strong evidence that μ=100 • There is strong evidence that μ≠100 • There is strong evidence that μ>100 • There is strong evidence that μ<100 STA 291 Summer 2010 Lecture 18

3. Example • The p-value for testing H1: µ≠100 is p=.001. In addition you know that the test statistic was z=3.29. This indicates that… • There is strong evidence that μ=100 • There is strong evidence that μ>100 • There is strong evidence that μ<100 STA 291 Summer 2010 Lecture 18

4. Rejection Region • Range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favor of the alternative hypothesis • Type of test determines which tail(s) the rejection region is in • Left-tailed • Right-tailed • Two-tailed STA 291 Summer 2010 Lecture 18

5. Test Statistic • Testing µ with n large • Just like finding a confidence interval for µ with n large • Reasons for choosing test statistics are the same as choosing the correct confidence interval formula STA 291 Summer 2010 Lecture 18

6. Test Statistic • Testing µ with n small • Just like finding a confidence interval for µ with n small • Reasons for choosing test statistics are the same as choosing the correct confidence interval formula • Note: It is difficult for us to find p-values for this test statistic because of the way our table is set up STA 291 Summer 2010 Lecture 18

7. Normality Assumption • An assumption for the t-test is that the population distribution is normal • In practice, it is impossible to be 100% sure if the population distribution is normal • It may be useful to look at histogram or stem-and-leaf plot (or normal probability plot) to check whether the normality assumption is reasonable • Good news • t-test is relatively robustagainst violations of this assumption • Unless the population distribution is highly skewed, the hypotheses tests and confidence intervals are valid • However, the random sampling assumption must never be violated, otherwise the test results are completely invalid STA 291 Summer 2010 Lecture 18

8. Example • A courier service advertises that its average delivery time is less than 6 hours for local deliveries. A random sample of times for 12 deliveries found a mean of 5.7 and a standard deviation of 1.58. Is this sufficient evidence to support the courier’s advertisement at α=.05? • State and test the hypotheses using the rejection region method. • Why wouldn’t the p-value method be good to use? STA 291 Summer 2010 Lecture 18

9. Correlation Between Tests and Confidence Intervals • 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) • Why does this make sense? • In general, a 100(1-α)% confidence interval corresponds to a test at significance level α STA 291 Summer 2010 Lecture 18

10. Example • A survey of 35 cars that just left their metered parking spaces produced a mean of 18 minutes remaining on the meter and a standard deviation of 22. Test the parking control officer’s claim that the average time left on meters is equal to 15 minutes. • State and test the hypotheses with a level of significance of 5% using the confidence interval method. STA 291 Summer 2010 Lecture 18