Exam 6: Inferences With Categorical Data

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. -If the test statistic turns out to be 12.7 , what can you say about the $p$ -value?

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Question 1

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If the test statistic turns out to be 12.7, then using Table IV with 3 degrees of freedom, you know that .005 < p-value < .01 .

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -Show how to calculate the expected count for the "conservative, men" cell.

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Question 2

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The expected count is $(1470 \times 1931)/4333 or 655.11$ .

Researchers studied whether hormone levels tend to differ following days of strenuous activity as opposed to days of no strenuous activity. For each of six subjects, one blood sample was taken after a day in which the person engaged in strenuous exercise, and one blood sample was taken after a day that included no strenuous activity. The data on growth hormone levels (in $\mathrm{mg} / \mathrm{ml}$ ) follow, along with some summary statistics: $Researchers studied whether hormone levels tend to differ following days of strenuous activity as opposed to days of no strenuous activity. For each of six subjects, one blood sample was taken after a day in which the person engaged in strenuous exercise, and one blood sample was taken after a day that included no strenuous activity. The data on growth hormone levels (in \mathrm{mg} / \mathrm{ml} ) follow, along with some summary statistics: a. Is a paired t -test or an independent-samples t -test appropriate here? Explain briefly. b. State the appropriate hypotheses, in words as well as in symbols. c. Calculate the test statistic, and determine the p -value as accurately as possible. d. Using the \alpha=.05 significance level, summarize the conclusion you would draw about this study. (Be sure to relate your conclusion to the context.) e. If the sample size were larger, and all else remained the same, would these results produce stronger, weaker, or the same level of evidence that hormone levels differ after strenuous activity? (No explanation is needed.)$ a. Is a paired $t$ -test or an independent-samples $t$ -test appropriate here? Explain briefly. b. State the appropriate hypotheses, in words as well as in symbols. c. Calculate the test statistic, and determine the $p$ -value as accurately as possible. d. Using the $\alpha=.05$ significance level, summarize the conclusion you would draw about this study. (Be sure to relate your conclusion to the context.) e. If the sample size were larger, and all else remained the same, would these results produce stronger, weaker, or the same level of evidence that hormone levels differ after strenuous activity? (No explanation is needed.)

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Question 3

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a. This study calls for a paired $t$ -test because the same six subjects were measured under both conditions.
b. $\mathrm{H}_{0}: \mu_{ diff}=0$ . The population mean of the differences in hormone levels, between days with strenuous activity and days without, is zero; there is no difference in population mean hormone levels depending on whether the subject had exercised strenuously or not.
$\mathrm{H}_{0}: \mu_{ diff} \neq 0$ . There is a difference in population mean hormone levels depending on whether the subject had exercised strenuously or not.
c. The test statistic is $t = \frac{\overline{x}d}{s_d/\sqrt{n_d}} = \frac{6.20}{7.53/\sqrt{6}} \approx 2.017$
Using the $t$ -table with $n-1=5$ degrees of freedom reveals the one-tailed probability to be between .025 and .05 , so the two-sided $p$ -value is between .05 and .10 .
d. Because the $p$ -value is greater than .05 , you fail to reject $\mathrm{H}_{0}$ at the .05 level. So, the data do not provide convincing evidence that the population mean hormone levels differ depending on whether the subject has engaged in strenuous exercise or not. But this does not mean you accept $\mathrm{H}_{0}$ and conclude that the population means are the same; it merely means you do not have enough evidence to conclude that they differ.
e. Larger sample sizes would provide stronger evidence of a difference. The test statistic would be larger and the $p$ -value smaller.

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. -If the $p$ -value turns out to be .004 , would you conclude that the four candies are equally likely or that they are not? Explain briefly.

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Question 4

Suppose a large-scale randomized experiment were conducted to compare two weight loss plans. At the end of the experiment, the weight loss for each subject (in pounds) was recorded. The $p$ -value turned out to be .004 , and a $99 \%$ confidence interval for the difference in means turned out to be $(0.26,0.84)$ . a. Do the data provide strong evidence that the mean weight loss differs between the two plans? Explain briefly. b. Do the data provide strong evidence that the mean weight loss differs considerably (i.e., a lot) between the two plans? Explain briefly. c. Describe briefly what a Type I error would mean in this context. d. Describe briefly what a Type II error would mean in this context.

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Question 5

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -Which cells of the table contribute the most to the calculation of the chi-square statistic? Explain what this reveals about the association between gender and political viewpoint.

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Question 6

A recent headline announced that mountain biking may reduce fertility in men. The article was based on a study that found that $90 \%$ of the avid mountain bikers in the study had low sperm counts, and $26 \%$ of the non-bikers in the study had low sperm counts. a. What further information would you need in order to do the calculations for the appropriate hypothesis test? b. What further information would you need in order to determine whether the technical conditions of this hypothesis test are satisfied? c. If the technical conditions are satisfied, and if the $p$ -value turns out to be very small, would you conclude that avid mountain biking causes lower sperm counts? Explain briefly.

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Question 7

A student recorded the number of calories in samples of vegetarian and nonvegetarian entrees served by a college dining hall. The student wanted to conduct a hypothesis test of whether the sample data provide evidence that the population mean calorie amounts differ between these two groups. Summary statistics of the data are provided in the following table: The two-sample $t$ -test statistic turns out to equal 0.92 . a. State the appropriate null and alternative hypotheses in symbols. b. Do you have enough information to check the technical conditions for a two-sample $t$ -test? Explain. c. Determine the $p$ -value as accurately as possible. (Explain what table you use and how you use it.) d. Summarize your conclusion in a sentence or two, as if you were the statistical consultant reporting back to the student who conducted the study. e. Describe how the $p$ -value would change (smaller, larger, or no change) if all else were the same except the following. (No explanations are needed.) The sample standard deviations were larger. The sample sizes were larger. The sample means were further apart.

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Question 8

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 - $p$ -value

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Question 9

"Low-carb" diets have become quite popular recently. In a recent study reported in the New England Journal of Medicine, researchers randomly assigned 63 obese men and women to either a low- carbohydrate diet or a conventional diet. Each subject's weight loss, as a percentage of body weight, was recorded after 3 months, after 6 months, and after 12 months. a. Is this an observational study or an experiment? Explain briefly. b. Identify the explanatory variable and the response variable. c. Some people exercise more than others, and exercise can affect weight loss. Explain why this exercise level is nevertheless not a confounding variable in this study. d. The researchers wanted to test the claim that people tend to lose more weight on low-carb diets than on conventional diets. Write the null and alternative hypotheses (in symbols only) for testing this claim. e. Minitab reports the $p$ -values to be .000 for the 3 -month weight losses, .008 for the 6 -month weight losses, and . 125 for the 12-month weight losses. Summarize the conclusions you would draw.

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Question 10

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 you need more information to calculate the test statistic? If so, specify what information. If not, calculate the test statistic.

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Question 11

In a study reported in the September 24, 2007, of Archives of Internal Medicine, German researchers described a study conducted on 1162 German adults suffering from chronic low-back pain. These subjects were randomly assigned to one of three groups: real acupuncture, sham acupuncture, or conventional therapy. Researchers found that $47 \%$ of subjects in the real acupuncture group improved compared to $44 \%$ in the sham acupuncture group and $27 \%$ in the conventional therapy group. a. What kind of test ( $z-$ , $t-$ , or chi-square) would be appropriate for testing whether the improvement proportions differ significantly among these three groups? (No explanation is needed.) b. Report the appropriate degrees of freedom for this test. c. The $p$ -value turns out to be very close to zero. Summarize the conclusion you would draw.

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Question 12

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. -State the appropriate null hypothesis in words.

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Question 13

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 you need more information to determine the degrees of freedom? If so, specify what information. If not, specify the appropriate value for the degrees of freedom. Suppose you take a random sample of candies and find 20% Dark Chocolate, 20% Regular Chocolate, $30 \%$ Krackel, and $30 \% \mathrm{Mr}$ . Goodbar.

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Question 14

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -State the appropriate null and alternative hypotheses for conducting a chi-square test on these data.

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Question 15

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. -Even if this were a random sample from the population of all students at the university, one of the technical conditions of the chi-square test would still not be satisfied with these data. Explain.

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Question 16

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 -Check of technical conditions

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Question 17

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -The $p$ -value of this test turns out to equal .004. Summarize the conclusion that you would draw from this test.

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Question 18

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. -If the $p$ -value turned out to be very small, what conclusion (in context) would you draw?

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Question 19

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 -Null and alternative hypotheses

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Question 20

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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. -If the test statistic turns out to be 12.7 , what can you say about the $p$ -value?

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -Show how to calculate the expected count for the "conservative, men" cell.

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -State the appropriate null and alternative hypotheses for conducting a chi-square test on these data.

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -The $p$ -value of this test turns out to equal .004. Summarize the conclusion that you would draw from this test.

Collecting Data and Drawing Conclusions

Summarizing Data

Randomness in Data

Inference From Data: Principles

Inference From Data: Comparisons

Relationships in Data

Analyzing Probabilities and Expected Values

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Exam 6: Inferences With Categorical Data

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. -If the test statistic turns out to be 12.7 , what can you say about the ppp -value?

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -Show how to calculate the expected count for the "conservative, men" cell.

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -State the appropriate null and alternative hypotheses for conducting a chi-square test on these data.

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -The ppp -value of this test turns out to equal .004. Summarize the conclusion that you would draw from this test.

Collecting Data and Drawing Conclusions

Summarizing Data

Randomness in Data

Inference From Data: Principles

Inference From Data: Comparisons

Relationships in Data

Analyzing Probabilities and Expected Values

Filters

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. -If the test statistic turns out to be 12.7 , what can you say about the $p$ -value?

The following table displays counts obtained from the 2006 General Social Survey; also presented in parentheses are expected counts: -The $p$ -value of this test turns out to equal .004. Summarize the conclusion that you would draw from this test.