Lecture 11

1. Lecture 11 Dan Piett STAT 211-019 West Virginia University

2. Last Week • Introduction to Hypothesis Testing • Hypothesis Tests for µ • Large Sample • Small Sample • Hypothesis Tests for p

3. Overview • Hypothesis Tests on a difference in means • Hypothesis Tests on a difference in proportions • The 2-sided alternative

4. Section 11.1 Hypothesis Tests on the Difference in Means

5. Difference in Means • Previously we created confidence intervals for the difference in two population means. • Male Scores vs Female Scores • This is the same idea we had when we did confidence intervals • Our same rules apply for determining large and small sample hypothesis tests

6. Large Sample Hyp. Test (n & m > 20) • H0: µx - µy = 0 (Does not have to be 0, but almost always is) • HA: µx - µy < 0 (µy is bigger) or µx - µy > 0 (µx is bigger) or µx - µy ≠ 0 • Alpha is .05 if not specified • Test Statistic = Z = • P-value will come from the normal dist. Table • For > alternative: P(z>Z) • For < alternative: P(z<Z) • For ≠ alternative:2*P(z>|Z|) • Our decision rule will be to reject H0 if p-value < alpha • We have (do not have) enough evidence at the .05 level to conclude that the mean of group x is ______ (<, >, ≠) the mean of group y Requires a large sample size for both groups and equal population standard deviations for both groups. Also requires independent random samples.

7. Example • A college statistics professor conjectures that students with good high school math backgrounds (2+ courses) perform better in a college statistics course than students with a poor high school math background (<2 courses). He randomly selects 35 students with a good math background and 45 students with a poor math background, and records exam scores from a college statistics course. Test the hypothesis that the mean score of the good background students will be higher than the mean score of the poor math background students. Use alpha = .10. The summary data is as follows:

8. Small Sample Hyp. Test (n or m < 20) • H0: µx - µy = 0 (Does not have to be 0, but almost always is) • HA: µx - µy < 0 (µy is bigger) or µx - µy > 0 (µx is bigger) or µx - µy ≠ 0 • Alpha is .05 if not specified • Test Statistic = T = • P-value will come from the t-dist. Table with df = n+m-2 • For > alternative: P(t>|T|) • For < alternative: P(t>|T|) • For ≠ alternative: 2*P(t>|T|) • Our decision rule will be to reject H0 if p-value < alpha • We have (do not have) enough evidence at the .05 level to conclude that the mean of group x is ______ (<, >, ≠) the mean of group y Requires both distributions are approximately normal with equal standard deviations. Also requires independent random samples.

9. Example • A researcher wishes to assess a “new” teaching method for “slow learners”. A random sample of 8 students use the new method, and a random sample of 12 students use the “standard” teaching method. After 6 months, an exam is administered to each student. Does the data indicate that the new teaching method is preferable? Use alpha = .05. The summary statistics are as follows:

10. Section 11.2 Hypothesis Tests for Two Independent Population Proportions

11. Difference in Pop. Proportions • We are again interested in the difference in the proportions of two populations • Proportion of A’s on Exam 1 vs. Proportion of A’s on Exam 2 • Much like all the other tests covered, the same rules apply in Hypothesis Testing that were involved in Confidence Intervals • Also we will only be considering the case where the above is true, therefore we will only be interested in tests using Z as the test statistic.

12. Hypothesis Tests on the difference of Proportions • H0: p1 – p2 = # (usually 0) • HA: p < # or p > # or p ≠ # • Alpha is .05 if not specified • Test Statistic = Z = • P-value will come from the normal dist. Table • For > alternative: P(z>Z) • For < alternative: P(z<Z) • For ≠ alternative:2*P(z>|Z|) • Our decision rule will be to reject H0 if p-value < alpha • We have (do not have) enough evidence at the .05 level to conclude that the proportion of group x is ______ (<, >, ≠) the proportion of group y Requires conditions on np’s. Also requires independent random samples

13. Examples • American Cancer Society wants to determine if the proportion of smokers in the population of Americans has decreased over the decade preceding 2002. In 1992, a random sample of 150 Americans showed 58 who smoked. In 2002, a random sample of 200 Americans included 64 who smoked. Does the data indicate that the proportion of smokers has decreased over the past decade? Use alpha = .05.

14. Section 11.3 The 2-sided alternative

15. Notes on 2 Sided Alternatives • Up until this point all of our examples have had alternative hypotheses of the form < or >. • What about ≠? • What we will do for this is take our previous p-values times 2 • We take the value that makes sense • If our statistic is less than our null hypothesis value, we use a < probability • If our statistic is more than our null hypothesis value, we use a > probability

16. Example • The quality control manager at a sugar processing packaging plant must make sure that two-pound bags of sugar actually contain two pounds of sugar. He randomly selects 50 bags of sugar and weighs their contents. The sample mean is 1.962 pounds with a sample std. dev of 0.160 pounds. Does this data indicate that the mean weight of all bags of sugar is different from 2 pounds? Use alpha = .05.