Dan PiettSTAT 211-019

West Virginia University

Lecture 11

Last WeekIntroduction to Hypothesis TestingHypothesis Tests for µ

Large SampleSmall Sample

Hypothesis Tests for p

OverviewHypothesis Tests on a difference in means Hypothesis Tests on a difference in

proportionsThe 2-sided alternative

Section 11.1

Hypothesis Tests on the Difference in Means

Difference in MeansPreviously 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. H0: µx - µy = 0 (Does not have to be 0, but almost always is)

2. HA: µx - µy < 0 (µy is bigger) or µx - µy > 0 (µx is bigger) or µx - µy ≠ 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 H0 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:

Group Mean Standard Deviation

Sample Size

2+ 84.2 10.2 35

<2 73.1 14.3 45

Small Sample Hyp. Test (n or m < 20)1. H0: µx - µy = 0 (Does not have to be 0, but almost always is)

2. HA: µx - µy < 0 (µy is bigger) or µx - µy > 0 (µx is bigger) or µx - µy ≠ 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 < alternative: P(t>|T|) For ≠ alternative: 2*P(t>|T|)

6. Our decision rule will be to reject H0 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.

ExampleA 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:

Group Mean Standard Deviation

Sample Size

New 77.125 4.853 8

Standard

72.333 6.344 12

Section 11.2

Hypothesis Tests for Two Independent Population Proportions

Difference in Pop. ProportionsWe are again interested in the difference in the

proportions of two populationsProportion of A’s on Exam 1 vs. Proportion of A’s

on Exam 2Much 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 Proportions1. H0: p1 – p2 = # (usually 0)

2. HA: p < # or 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 H0 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

ExamplesAmerican 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.

Section 11.3

The 2-sided alternative

Notes on 2 Sided AlternativesUp 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 2We 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

ExampleThe 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.