T-Tests for Comparing Two Means

1. T-Tests for Comparing Two Means Presentation 8.6

2. Notation Reminder for Comparing Two Means

3. Assumptions and Degrees of Freedom • The assumptions for the significance test are the same as they were for the confidence interval. • These must be explicitly addressed! • The degrees of freedom (df), for the test is also the same as it was for the confidence interval. • The calculations for the two-sample test should be handled by the calculator. • That’s why it’s called a calculator =)

4. Two-Sample t Test for Comparing Two Population Means If n1 and n2 are both large or if the population distributions are normal and when the two random samples are independently selected, the test statistic is:

5. Alternate hypothesis and finding the P-value: • Ha: µ1 - µ2 > 0 (or something else) P-value = Area under the t distribution to the right of the calculated t • Ha: µ1 - µ2 < 0 (or something else) P-value = Area under the t distribution to the left of the calculated t Hypotheses for Two-Sample t Tests(Difference of Two Means)

6. Ha: µ1 - µ2 0 (or something else) • 2•(area to the right of t) if t is positive • 2•(area to the left of t) if t is negative Hypotheses for Two-Sample t Tests(Difference of Two Means)

7. Example #1 • A extermination company has developed a new pesticide that may kill cockroaches more quickly. They conduct an experiment and hope to show the new pesticide does indeed work faster.

8. Example #1 • The data is the number of days it took to eliminate a contained colony of cockroaches. That is, x-bar is the mean number of days it took for all the cockroaches in the colony to die.

9. Example #1 m1 = the mean number of days to kill the cockroaches using the old pesticide m2 = the mean number of days to kill the cockroaches using the new pesticide H0:m1 = m2 or (m1 - m2 = 0) Ha:m1>m2 or (m1 - m2>0) Significance level:a = 0.05 Since the sample sizes are large, the t procedures are appropriate.

10. Example #1 • Conduct Calculations • I would suggest writing the statistics into the formula and then letting the calculator calculate.

11. Example #1 • Conclusions • With p=.0514, this is greater than .05, therefore we fail to reject the null. • There is not sufficient evidence to suggest that the new pesticide works faster than the old.

12. Example #2 In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. Use a significance level of 0.01. Brand ABrand B 517, 495, 503, 491 493, 508, 513, 521 503, 493, 505, 495 541, 533, 500, 515 498, 481, 499, 494 536, 498, 515, 515 State and perform an appropriate hypothesis test.

13. Example #2 m1 = the mean amount of acetaminophen in cold tablet brand A m2 = the mean amount of acetaminophen in cold tablet brand B H0:m1 = m2 (m1 - m2 = 0) Ha:m1m2(m1 - m20) Significance level:a = 0.01

14. Example #2 Assumptions:The samples were selected independently and randomly. Since the samples are small, we check for normality using normal probability plots.

15. Example #2 Assumptions (continued): As we can see from the normal probability plots and the boxplots above, the assumption that the underlying distributions are normally distributed is reasonable.

16. Example #2 • Conduct Calculations

17. Example #2 • Conclusions • With p=.0025, this is less than .01, therefore we reject the null. • There is sufficient evidence to suggest that the two brands of cold medicine contain different amounts of acetaminophen.

18. T-Tests for Comparing Two Means • This concludes this presentation.