Inferences About Proportions

testing exercises


Questions

1. A civil rights group feels that the selection of contestants on a game show program is biased against blacks. The U.S. Census Bureau reports that 12.7% of all U.S. citizens are black. Suppose we can assume that the contestants who have already appeared may be treated as essentially a random sample drawn from all contestants who may ever appear. Of these appearing contestants there have been 212 whites (nonHispanic), 13 blacks, 17 Hispanics, and 3 Asians. Test the groups claim: state hypotheses, compute the test statistic and obtain the P-value. Interpret the P-value. Is there sufficient evidence to conclude that the civil rights group is correct? Use a 1% significance level to make a decision. In practical terms, what issues must be considered before accusing the game show of discrimination?

2. According to the U.S. Census Bureau's most recent reported statistics (1998), what percentage of U.S. citizens fall into each of the following groups? (I've already given you one figure above.)

White non Hispanic ______ %
Black ______ %
Asian & Pacific Islander ______ %
American Indian, Eskimo, Aleut      ______ %
Hispanic ______ %

What percentage are female?

3. In a Pepsi ad, the claim is made that "in recent blind taste tests, more than half the Coke drinkers said they preferred the taste of Pepsi." In this test, 100 Coke drinkers took the "Pepsi Challenge" and 57 prefered the taste of Pepsi. Use this data to test the alternative hypothesis that more than half of ALL Coke drinkers will prefer Pepsi in a blind taste test: state hypotheses, compute the test statistic and obtain the P-value. Interpret the P-value. If testing at the 5% level, what do you conclude? Suppose 65 preferred Pepsi. Without computing: Would the P-value be larger, smaller, or the same as what you've just obtained? Now, compute it and see if you're right.


Solutions

1. We're testing H0: p = .127    HA: p < .127  where p is the proportion of all game show contestants who are black. The test statistic is Z = -3.48 and the P-value is .0003. When the null hypothesis is true there is a 1 in 3333 (=.0003) chance of observing data such as ours (only 13 of 245, or 5.31% black). Inference: the null hypothesis probably is not true. At that 1% significance level we reject the null and conclude that the blacks have less that 12.7% chance of participating on the show.

2. Find out for yourself! It's on the net. Try the U.S. Census Bureau.

3. We're testing H0: p = .5    HA: p > .5  where p is the proportion of all Coke drinkers who prefer Pepsi when tested. The test statistic is Z = 1.40 and the P-value is .0808. When the null hypothesis is true there is about a 1 in 12 chance of observing data such as ours (the 57%). Inference: there's some evidence that the null hypothesis is not true -- but it's hardly compelling. At that 5% significance level we are unable to reject the null hypothesis. If 65 out of 100 had preferred Pepsi we'd have more evidence against the null hypothesis -- consequentlly the P-value would be smaller. It would be .0013 or 1 in approximately 770.