Dan Piett STAT West Virginia University Lecture 11

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

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

Section 11.1 Hypothesis Tests on the Difference in Means

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

Large Sample Hyp. Test (n & m > 20) 1. H 0 : µ x - µ y = 0 (Does not have to be 0, but almost always is) 2. H A : µ x - µ y 0 (µ x is bigger) or µ x - µ y ≠ 0 3. Alpha is.05 if not specified 4. Test Statistic = Z = 5. 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|) 6. Our decision rule will be to reject H 0 if p-value < alpha 7. 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.

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: GroupMeanStandard Deviation Sample Size <

Small Sample Hyp. Test (n or m < 20) 1. H 0 : µ x - µ y = 0 (Does not have to be 0, but almost always is) 2. H A : µ x - µ y 0 (µ x is bigger) or µ x - µ y ≠ 0 3. Alpha is.05 if not specified 4. Test Statistic = T = 5. P-value will come from the t-dist. Table with df = n+m-2 For > alternative: P(t>|T|) For |T|) For ≠ alternative: 2*P(t>|T|) 6. Our decision rule will be to reject H 0 if p-value < alpha 7. 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.

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: GroupMeanStandard Deviation Sample Size New Standard

Section 11.2 Hypothesis Tests for Two Independent Population Proportions

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.

Hypothesis Tests on the difference of Proportions 1. H 0 : p 1 – p 2 = # (usually 0) 2. H A : p # or p ≠ # 3. Alpha is.05 if not specified 4. Test Statistic = Z = 5. 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|) 6. Our decision rule will be to reject H 0 if p-value < alpha 7. 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

Examples American Cancer Society wants to determine if the proportion of smokers in the population of Americans has decreased over the decade preceding 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.

Section 11.3 The 2-sided alternative

Notes on 2 Sided Alternatives Up until this point all of our examples have had alternative hypotheses of the form. 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

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 pounds with a sample std. dev of pounds. Does this data indicate that the mean weight of all bags of sugar is different from 2 pounds? Use alpha =.05.