Kenwood Academy High School



Test 10 AP Statistics Name:________________________

Part 1: Multiple Choice. Circle the letter corresponding to the best answer.

1. A city planner is comparing traffic patterns at two different intersections. He randomly selects 12 times between 6 am and 10 pm, and he and his assistant count the number of cars passing through each intersection during the 10-minute interval that begins at that time. He plans to test the hypothesis that the mean difference in the number of cars passing through the two intersections during each of those 12 times intervals is 0. Which of the following is appropriate test of the city planner’s hypothesis?

(a) Two-proportion z-test

(b) Two-sample z-test

(c) Matched pairs t-test

(d) Two proportion t-test

(e) Two-sample t-test

Use the following for questions 2 and 3

Different varieties of fruits and vegetables have different amounts of nutrients. These differences are important when these products are used to make baby food. We wish to compare the carbohydrate content of two varieties of peaches. The data were analyzed with MINITAB, and the following output was obtained:

2. We wish to test if the two varieties are significantly different in their mean carbohydrate content. Which of the following are the appropriate null and alternative hypotheses for this situation?

[pic]

3. Assuming the conditions for inference were met, which of the following is the most appropriate conclusion to draw at the ( = 0.05 level?

(a) The test provides convincing evidence that the carbohydrate content of variety 1 is higher than variety 2.

(b) The test provides convincing evidence that the carbohydrate contents of the two varieties are equal.

(c) We accept Ha: variety 1 has a higher carbohydrate content than variety 2.

(d) We reject H0: variety 1 has a higher carbohydrate content than variety 2.

(e) We cannot reject H0: we do not have convincing evidence that the carbohydrate contents are different.

4. Popular wisdom is that eating presweetened cereal tends to increase the number of dental caries (cavities) in children. A sample of children was (with parental consent) entered into a study and followed for several years. Each child was classified as a sweetened-cereal lover or a unsweetened cereal lover. At the end of the study, the amount of tooth damage was measured. Here are the summary data:

[pic]Assuming the necessary conditions for inference are met, which of the following is an approximate 95% confidence interval for the difference in the mean tooth damage?

5. You are constructing a 90% confidence interval for the difference of means from simple random samples from two independent populations. The sample sizes are [pic] and [pic] . You draw dot plots of the samples to check the normality condition for two-sample t-procedures. Which of the following descriptions of those dot plots would suggest that it is safe to use t-procedures?

I. The dot plot of sample 1 is roughly symmetric, while the dot plot of sample 2 is moderately skewed left. There are no

outliers.

II. Both dot plots are roughly symmetric. Sample 2 has an outlier.

III. Both dot plots are strongly skewed to the right. There are no outliers.

(a) I only

(b) II only

(c) I and II

(d) I, II, and III

(e) t-procedures are not recommended in any of these cases.

Use the following for questions 6 – 8:

Janice and her cousin Linda are a little competitive about the relative merits of their home towns. One contest they had was to determine who had more rainy days. They found weather records on the internet and each of them randomly selected 60 days from the past 5 years. Janice found that there had been measurable rainfall on 17 of the 60 days she selected for Asheville, and Linda found that there had been measurable rainfall on 12 of the 60 days she selected for Lincoln. They intend to perform a test of significance on their data, using the hypotheses [pic] versus[pic] and the 0.05 significance level.

6. When calculating the test statistic, what expression would they use to estimate the standard deviation of the sampling distribution of the difference in proportions, [pic] ?

7. Janice and Linda’s test statistic is 1.07. Which of the following is closest to the appropriate P-value for the test?

(a) 0.0446 (b) 0.0892 (c) 0.1423 (d) 0.1449 (e) 0.2846

8. Which of the following best describes what it would mean if Janice and Linda’s test resulted in a Type I error?

(a) Concluding that there is a difference in the proportion of rainy days in the two cities when there is no difference.

(b) Concluding that there is no difference in the proportion of rainy days in the two cities when there is a difference.

(c) Choosing the wrong test procedure, such as using a z-test instead of a t-test.

(d) Accepting the alternative hypothesis instead of rejecting the null hypothesis.

(e) Accepting the null hypothesis instead of rejecting the alternative hypothesis.

Use the following for questions 9 – 11:

Sixty-eight people from a random sample of 128 residents of Uppsala, Sweden, had blue eyes. 45 people from a random sample of 110 people from Preston, England, had blue eyes. Let [pic]represent the proportion of people in Uppsala with blue eyes, and let [pic]represent the proportion of people in Preston with blue eyes.

9. If researchers suspected that the distribution of eye color is different in these two countries before collecting the data, which of the following pairs of hypotheses would be appropriate to test?

[pic]

10. Which of the following conditions are necessary in order to perform the test in Question 1?

I. There must be at least 1280 people in Uppsala, Sweden and at least 1100 people in Preston, England.

II. np and n(1 – p) must be large enough for Normal calculations to be reasonably accurate.

III. Two independent random samples must be taken.

(a) I only

(b) II only

(c) III only

(d) I and III are necessary

(e) I, II, and III are all necessary

11. The P-value for this test is 0.06. If the researchers chose a significance level of 0.05, which of the following represents the correct conclusion to draw from this result?

(a) Reject [pic]there is evidence to suggest a difference in the proportion of blue-eyed people in these two countries.

(b) Fail to reject [pic]there is convincing evidence to suggest a difference in the proportion of blue-eyed people in these two

countries.

(c) Fail to reject [pic]there is insufficient evidence to suggest a difference in the proportion of blue-eyed people in these two

countries.

(d) Accept Ha: there is convincing evidence to suggest a difference in the proportion of blue-eyed people in these two

countries.

(e) Accept [pic]there is insufficient evidence to suggest a difference in the proportion of blue-eyed people in these two

countries.

12. The following are percents of fat found in 5 samples of each of two brands of ice cream:

[pic]

Which of the following procedures is appropriate to test the hypothesis of equal average fat content in the two types of ice cream?

(a) Paired t test with 5 df.

(b) Two-sample t test with 4 df.

(c) Paired t test with 4 df.

(d) Two-sample t test with 9 df.

(e) Two-proportion z test

13. An ecologist studying differences in populations of a certain species of lizards on two different islands collects lizards in live traps, weighs them, and then releases them again. (He marks them so he won’t weigh the same lizard twice). During one study period, he collects the following data. All weights are in grams.

[pic]

Which of the following is the correct expression for the test statistic to test the hypothesis that the mean weights on the two islands are equal?

Use the following for questions 14 – 16:

An experiment was conducted to assess the efficacy of spraying oats with malathion (at 0.25 lb/acre) to control the cereal leaf beetle. A sample of 10 farms was selected at random from southwest Manitoba. Each farm was assigned at random to either the control group (no spray) or the treatment group (spray). At the conclusion of the experiment, a plot on each farm was selected, the number of larvae per stem was measured, and a one-tailed test of significance was performed to determine if malathion reduced the number of beetles. Here are two possible outputs from MINITAB, only one of which is correct (some output hidden):

[pic]

14. Which of the following is the appropriate test statistic and a possible P-value?

(a) 1.896, 0.013 (b) 1.896, 0.065 (c) 1.896, 0.131 (d) 1.887, 0.059 (e) 1.887, 0.118

15. In which one of the following cases would a Type II error occur?

(a) We do not conclude malathion is effective when in fact it was effective.

(b) We conclude malathion is effective when in fact it is ineffective.

(c) We conclude malathion is effective when in fact it is effective.

(d) We do not conclude malathion is effective when in fact it is ineffective.

(e) We conclude malathion is neither ineffective nor effective.

16. What does power refer to in this situation?

(a) The ability to detect an effect of malathion when in fact there is no effect.

(b) The ability to not detect an effect of malathion when in fact there is no effect.

(c) The ability to detect an effect of malathion when in fact there is an effect.

(d) The ability to not detect an effect of malathion when in fact there is an effect.

(e) The ability to generalize the results of this controlled experiment to other insect pests besides the cereal leaf beetle.

Part 2: Free Response

Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations.

9. Nicotine patches are often used to help smokers quit. Does giving medicine to fight depression also help? A randomized double-blind experiment assigned 244 smokers to receive nicotine patches and another 245 to receive both a patch and the antidepressant drug

bupropion. After a year, 40 subjects in the nicotine patch group had abstained from smoking, as had 87 in the patch-plus-drug group.

(a) Construct and interpret a 99% confidence interval for the difference in the proportion of smokers who abstain when using buproprion and a nicotine patch and the proportion who abstain when using only a patch.

(b) Based only on this interval, do you think that the difference in proportion of abstaining smokers is significant? Justify your answer.

10. A study of iron deficiency among infants compared blood hemoglobin levels of a random sample of one-year-old infants who had been breast-fed to a random sample of one-year old infants who had been fed with standard infant formula. Here are the results.

[pic]

We wish to test the hypothesis [pic]against [pic]where [pic]and [pic]are the population mean blood hemoglobin levels for breast-fed and formula-fed infants, respectively.

(a) What additional information would you need to confirm that the conditions for this test have been met?

(b) Assuming the conditions have been met, calculate the test statistic and P-value for this test.

(c) Interpret the P-value in the context of this study, and draw the appropriate conclusion at the ( = 0.05 level.

(d) Given your conclusion in part (c), which type of error, Type I or Type II, is it possible to make? Describe that error in the context of this study.

11. We all know that regular exercise (combined with a sensible diet) is a key to shedding those extra pounds. Experience shows that overweight people find it tough to keep exercising. Perhaps they will do better with several short sessions each day rather than one longer session. Perhaps having exercise equipment at home will help. An experiment looked at these issues. The subjects were women aged 25 to 45 whose weights were 20% to 75% higher than ideal. The study report says: Subjects were randomly assigned to 1 of 3 groups. All groups were prescribed a similar volume of exercise. The 3 groups differed in the way the exercise was prescribed.

Short-Bout Exercise Group Fifty-one subjects were instructed to exercise 5 days/wk . . .However, rather than exercising continuously for the prescribed duration, subjects were instructed to divide the exercise into multiple 10-minute bouts that were performed at convenient times throughout the day.

Short-Bout plus Exercise Equipment Group The exercise prescription was identical to the exercise prescribed for the short-bout group. The 48 subjects in this group were also provided with motorized home treadmills.

Long-Bout Exercise Group Forty-nine subjects were instructed to exercise 5 days/wk; duration progressed from 20 min/day to 40 min/day. Participants performed the exercise in one long bout.

The researchers recorded weight, fitness, and whether the subject continued the exercise program.

a. Use a diagram to outline the design of this experiment.

b. How many subjects are there in all? Use Table D, starting at line 114, to assign the first 10 subjects for the long-bout group.

c. Could this study have been conducted in a double-blind manner? Explain your answer.

Here is a small part of the summary of the results:

There was no significant difference between the LB and SB groups for mean [SD] weight loss at 18 months

(LB, – 5.8 [7.1] kg; SB, – 3.7 [6.6] kg).

It gives the mean and standard deviation of the weight loss (in kilograms) for the long-bout (LB) and short-bout (SB) groups. The data come from the 37 LB subjects and 36 SB subjects who completed the study.

d. Do an appropriate test to confirm the report that there is not a significant difference in the mean weight loss in the two groups.

e. Construct and interpret a 95% confidence interval for the difference in mean weight loss for similar subjects using these two treatments. Does the confidence interval confirm that there is no significant difference in mean weight loss?

f. In many studies, subjects drop out before the study is complete. Suppose that no one dropped out of the long-bout or short-bout groups and that the means and standard deviations for both groups remained the same. Would this change your decision from question 4? Explain.

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