Exam 6: Inferences With Categorical Data

For each of the following situations, indicate whether an independent-samples analysis or a matchedpairs analysis would be appropriate. (Do not bother to explain.) a. You want to investigate whether cows milked by a mechanical device tend to produce more or less milk than cows milked by hand. You have 20 cows to work with, so you randomly assign half of them to be milked by machine and half by hand. b. You want to investigate whether there is a difference in the average age or cars driven by students and faculty at your university, so you randomly select 30 students and 10 faculty members and ask them to report the age of the cars that they drive. c. You want to investigate whether college freshmen tend to gain 15 pounds of weight, as the "freshman fifteen" legend suggests. You recruit 40 freshmen to participate in your study. You weigh them at the beginning of the fall term and again at the end of the fall term.

On June 4-24, 2007, the Gallup Poll asked a random sample of adult Americans about their attitudes toward interracial marriage. Of the 2388 adults interviewed, $77 \%$ said they approve of marriage between whites and blacks. In a similar study conducted five years earlier in 2002, 65\% said they approve of such a marriage. (Assume that the sample size in 2002 was the same as in 2007.) a. Do these studies make use of random sampling or random assignment? (No explanation is necessary.) b. Determine a $95 \%$ confidence interval for the difference in population proportions who approve of such marriages between 2002 and 2007. c. Comment on what you can discern from this interval about whether, and if so how, this proportion has changed over that five-year period.

We have been told that $45 \%$ of all Reese's Pieces are orange, whereas $27.5 \%$ are brown and $27.5 \%$ are yellow. To investigate this hypothesis, some students counted the number of candies of each color in a bag of 500 Reese's Pieces. They found 230 orange candies, 138 brown candies, and 132 yellow candies. Perform the appropriate test by stating the hypotheses, calculating the test statistic and $p$ -value, and indicating what decision you would make (in context) using the $\alpha=.10$ significance level.

Students in a statistics class were asked which of three lifetime achievements they would most like to win: a Nobel Prize, an Academy Award, or an Olympic Gold Medal. Results were that 17 students chose the Nobel Prize, 10 chose the Academy Award, and 18 chose the Olympic Gold Medal. Consider these students as a random sample of students at their university. Conduct a chi-square goodness-of-fit test of whether these sample data provide evidence to believe that the three choices are not equally likely in the population. Be sure to report and show the details of your calculations for -Expected counts and test statistic

Suppose 5000 students are randomly assigned to either take an SAT coaching course or not, with the following results in their improvements in SAT scores: A $99 \%$ confidence interval for the difference in population means turns out to be $(0.72,2.88)$ . a. Based on this interval, would you reject the null hypothesis that the population means are equal at the .01 significance level? (No explanation is needed.) b. Which is the more appropriate conclusion to draw: (i) that the data provide very strong evidence that the population means differ, or (ii) that the data provide evidence that the population means differ substantially? (No explanation is needed.)

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -The following values are used in calculating the test statistic. The one missing value is for the "conservative, men" cell. Show how to calculate this value. $X^{2} = 0.338+0.420+1.391+1.731+3.179+\_\_\_\_\_$

Hershey offers a sampler bag that contains four different kinds of candy bars. You want to test whether the random process that fills the bag with the different candies makes them each equally likely. -Would your test use the normal distribution, $t$ -distribution, or chi-square distribution?

Students in a statistics class were asked which of three lifetime achievements they would most like to win: a Nobel Prize, an Academy Award, or an Olympic Gold Medal. Results were that 17 students chose the Nobel Prize, 10 chose the Academy Award, and 18 chose the Olympic Gold Medal. Consider these students as a random sample of students at their university. Conduct a chi-square goodness-of-fit test of whether these sample data provide evidence to believe that the three choices are not equally likely in the population. Be sure to report and show the details of your calculations for -Conclusion in context

a. What's wrong with expressing a null hypothesis as $\mathrm{H}_{0} : \overline{x}_A = \overline{x}_B$ ? b. Suppose you take a random sample of 1000 Californians and an independent random sample of 1000 Pennsylvanians and measure the IQ of each person. Suppose that a 99% confidence interval for the difference in population means turns out to be $(-0.62,1.94)$ . Are the sample results statistically significant at the $alpha=.01$ level? Explain briefly.

A study of drinking habits of college students produced the following table of counts: a. Is this an observational or an experimental study? Explain briefly. b. What proportion of students who live with their parents say that they abstain from drinking alcohol? Consider the following computer output for a chi-square test: c. Show how to calculate the expected count for the "abstain from drinking/ reside off-campus with parents" cell of the table. d. What conclusion would you draw from this chi-square test? Explain briefly. e. Which cell contributes the most to the calculation of the chi-square statistic? What does this reveal about a possible relationship between a college student's location of residence and drinking habits? Explain.

Students in a statistics class were asked which of three lifetime achievements they would most like to win: a Nobel Prize, an Academy Award, or an Olympic Gold Medal. They were also asked to indicate their gender. Results are shown in the following table: Suppose that you were to conduct a chi-square test on these data. -Determine the expected count for women who would choose a Nobel Prize.

Students in a statistics class were asked which of three lifetime achievements they would most like to win: a Nobel Prize, an Academy Award, or an Olympic Gold Medal. They were also asked to indicate their gender. Results are shown in the following table: Suppose that you were to conduct a chi-square test on these data. -Report the value that you would you use for degrees of freedom with this test.