dan piett stat 211-019 west virginia university lecture 11
TRANSCRIPT
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
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
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.
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.