advertisement

Test 11A AP Statistics Name: Directions: Work on these sheets. Tables and formulas appear on a separate sheet. Part 1: Multiple Choice. Circle the letter corresponding to the best answer 1. DDT is an insecticide that accumulates up the food chain. Predator birds can be contaminated with quite high levels of the chemical by eating many lightly contaminated prey. One effect of DDT upon birds is to inhibit the production of the enzyme carbonic anhydrase, which controls calcium metabolism. It is believed that this causes eggshells to be thinner and weaker than normal and makes the eggs more prone to breakage. (This is one of the reasons why the condor in California is near extinction.) An experiment was conducted where 16 sparrow hawks were fed a mixture of 3 ppm dieldrin and 15 ppm DDT (a combination often found in contaminated prey). The first egg laid by each bird was measured, and the mean shell thickness was found to be 0.19 mm. A “normal” eggshell has a mean thickness of 0.2 mm. The null and alternative hypotheses are (a) H 0 : 0.2; H a : 0.2 (b) H 0 : 0.2; H a : 0.2 (c) H 0 : x 0.2; H a : x 0.2 (d) H 0 : x 0.19; H a : x 0.2 (e) H 0 : 0.2; H a : 0.2 2. A significance test allows you to reject a hypothesis H 0 in favor of an alternative Ha at the 5% level of significance. What can you say about significance at the 1% level? (a) H 0 can be rejected at the 1% level of significance. (b) There is insufficient evidence to reject H 0 at the 1% level of significance. (c) There is sufficient evidence to accept H 0 at the 1% level of significance. (d) Ha can be rejected at the 1% level of significance. (e) The answer can’t be determined from the information given. 3. In a test of H0: µ = 100 against Ha: µ 100, a sample of size 80 produces z = 0.8 for the value of the test statistic. The P-value of the test is thus equal to (a) 0.20 (b) 0.40 (c) 0.29 (d) 0.42 (e) 0.21 5. A 95% confidence interval for µ is calculated to be (1.7, 3.5). It is now decided to test the hypothesis H0: µ = 0 versus Ha: µ 0 at the = 0.05 level, using the same data as used to construct the confidence interval. (a) We cannot test the hypothesis without the original data. (b) We cannot test the hypothesis at the = 0.05 level since the = 0.05 test is connected to the 97.5% confidence interval. (c) We can make the connection between hypothesis tests and confidence intervals only if the sample sizes are large. (d) We would reject H0 at level = 0.05. (e) We would accept H0 at level = 0.05. Chapter 11 1 Test 11A 6. Here's a quote from a medical journal: “An uncontrolled experiment in 17 women found a significantly improved mean clinical symptom score after treatment. Methodologic flaws make it difficult to interpret the results of this study.” The authors of this paper are skeptical about the significant improvement because (a) there is no control group, so the improvement might be due to the placebo effect or to the fact that many medical conditions improve over time. (b) the P-value given was P = 0.03, which is too large to be convincing. (c) the response variable might not have an exactly Normal distribution in the population. (d) the study didn’t use enough subjects to achieve any statistically significant findings. (e) the mean is not resistant. 7. A medical experiment compared the herb echinacea with a placebo for preventing colds. One response variable was “volume of nasal secretions” (if you have a cold, you blow your nose a lot). Take the average volume of nasal secretions in people without colds to be = 1. An increase to = 3 indicates a cold. The significance level of a test of H 0 : 1 versus H a : 1 is (a) the probability that the test rejects H 0 when = 1 is true. (b) the probability that the test rejects H 0 when = 3 is true. (c) the probability that the test fails to reject H 0 when = 3 is true. (d) the probability that the test fails to reject H 0 when = 1 is true. (e) none of the above 8. A radio show runs a phone-in survey each morning. One morning the show asked its listeners whether they would prefer Congress or the president to set policy for the nation. The majority of those phoning in their responses answered “Congress,” and the station reported the results as statistically significant. We may safely conclude that (a) there is deep discontent in the nation with the president. (b) it is unlikely that, if all Americans were asked their opinion, the result would differ from that obtained in the poll. (c) there is strong evidence that the majority of Americans prefer Congress to set national policy. (d) very few people other than the majority of those phoning in their responses prefer Congress to set policy for the nation. (e) that the majority of Americans would actually prefer the president to set policy, because of the biased method of data collection. PART II: 1. The average time it takes for a person to experience pain relief from aspirin is 25 minutes. A new ingredient is added to help speed up relief. Let µ denote the average time to obtain pain relief with the new product. An experiment is conducted to verify if the new product is better. What are the null and alternative hypotheses? (a) H0 : µ = 25 vs. Ha : µ 25 (b) H0 : µ = 25 vs. Ha : µ < 25 (c) H0 : µ < 25 vs. Ha : µ = 25 (d) H0 : µ < 25 vs. Ha : µ > 25 (e) H0 : µ = 25 vs. Ha : µ > 25 2. A significance test was performed to test the null hypothesis H 0 : µ = 2 versus the alternative Chapter 11 2 Test 11A Ha: µ > 2. The test statistic is z = 1.40. The P-value for this test is approximately (a) 0.16 (b) 0.08 (c) 0.003 (d) 0.92 (e) 0.70 (f) None of the above. The answer is . 3. In a test of H0: µ = 100 against Ha: µ 100, a sample of size 10 produces a sample mean of 103 and a P-value of 0.08. Thus, at the 0.05 level of significance (a) there is sufficient evidence to conclude that µ 100. (b) there is sufficient evidence to conclude that µ = 100. (c) there is insufficient evidence to conclude that µ = 100. (d) there is insufficient evidence to conclude that µ 100. (e) there is sufficient evidence to conclude that µ = 103. 5. A certain population follows a Normal distribution with mean and standard deviation = 2.5. You collect data and test the hypotheses H 0 : = 1, H a : 1 You obtain a P-value of 0.022. Which of the following is true? (a) A 95% confidence interval for will include the value 1. (b) A 95% confidence interval for will include the value 0. (c) A 99% confidence interval for will include the value 1. (d) A 99% confidence interval for will include the value 0. (e) None of these is necessarily true. 6. Vigorous exercise helps people live several years longer (on the average). Whether mild activities like slow walking extend life is not clear. Suppose that the added life expectancy from regular slow walking is just 2 months. A statistical test is more likely to find a significant increase in mean life if (a) it is based on a very large random sample and a 5% significance level is used. (b) it is based on a very large random sample and a 1% significance level is used. (c) it is based on a very small random sample and a 5% significance level is used. (d) it is based on a very small random sample and a 1% significance level is used. (e) The size of the sample doesn't have any effect on the significance of the test. 7. Which of the following is not a condition for performing inference about a population mean ? (a) Inference is based on n independent measurements. (b) The population distribution is Normal or the sample size is large (say n > 30). (c) To use a z test, we must know the population standard deviation . (d) The data are obtained from an SRS from the population of interest. (e) Both np and n(1 – p) are 10 or greater. 8. Does taking ginkgo tablets twice a day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who took ginkgo tablets twice a day for a period of six months. At the end of the study, 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Nine of these variables were Chapter 11 3 Test 11A significantly better (in the sense of statistical significance) at the = 0.05 level for the group taking the ginkgo tablets as compared to the population as a whole, and one variable was significantly better at the = 0.01 level for the group taking the ginkgo tablets as compared to the population as a whole. It would be correct to conclude that (a) there is very good statistical evidence that taking ginkgo tablets twice a day provides some improvement in mental performance. (b) there is very good statistical evidence that taking ginkgo tablets twice a day provides improvement for the variable that was significant at the = 0.01 level. We should be cautious about making claims for the variables that were significant at the = 0.05 level. (c) these results would have provided very good statistical evidence that taking ginkgo tablets twice a day provides some improvement in mental performance if the number of subjects had been larger. It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured. (d) there is very good statistical evidence that taking ginkgo tablets twice a day provides significant improvement in ten specific areas related to mental condition. (e) none of the above PART III: 1. An opinion poll asks a random sample of adults whether they favor banning ownership of handguns by private citizens. A commentator believes that more than half of all adults favor such a ban. The null and alternative hypotheses you would use to test this claim are ˆ 0.5; H a : pˆ 0.5 (a) H 0 : p (b) H 0 : pˆ 0.5; H a : pˆ 0.5 (c) H 0 : p 0.5; H a : p 0.5 (d) H 0 : p 0; H a : p 0 (e) None of the above. The answer is _____________________________. 2. Bags of a certain brand of tortilla chips claim to have a net weight of 14 ounces. Net weights actually vary slightly from bag to bag and are Normally distributed with mean . A representative of a consumer advocate group wishes to see if there is any evidence that the mean net weight is less than advertised and so intends to test the hypotheses H0: = 14, Ha: < 14 To do this, he selects 16 bags of this brand at random and determines the net weight of each. He finds the sample mean to be x = 13.82 and the sample standard deviation to be s = 0.24. We conclude that we would (a) reject H0 at significance level 0.10 but not at 0.05. (b) reject H0 at significance level 0.05 but not at 0.025. (c) reject H0 at significance level 0.025 but not at 0.01. (d) reject H0 at significance level 0.01. (e) fail to reject H0 at the = 0.10 level. 4. You are thinking of using a t procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect that the distribution of the population is not Normal and may be moderately skewed. Which of the following statements is correct? (a) You should not use the t procedure because the population does not have a Normal distribution. (b) You may use the t procedure if your sample size is large, say, at least 50. (c) You may use the t procedure, but you should probably claim only that the significance level is 0.10. (d) You may not use the t procedure. The t procedures are robust to non-Normality for confidence intervals but not for tests of hypotheses. (e) You may use the t procedure if there are no outliers. Chapter 11 4 Test 11A 5. After once again losing a football game to the archrival, a college’s alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was taken. 64 of the alumni in the sample were in favor of firing the coach. Suppose you wish to see if a majority of living alumni are in favor of firing the coach. The appropriate test statistic is (a) z (0.64 0.5) (0.64)(0.36) 100 (b) z (0.64 0.5) (0.5)(0.5) 100 (c) z (0.64 0.5) (0.64)(0.36) 64 (d) z (0.64 0.5) (0.5)(0.5) 64 (e) t (0.64 0.5) (0.5)(0.64) 100 6. We prefer the t procedures to the z procedures for inference about a population mean because (a) z can be used only for large samples. (b) z requires that you know the population standard deviation . (c) z requires that you can regard your data as an SRS from the population of interest. (d) z requires that your population be Normally distributed. (e) z requires that your observations be independent. 7. Looking online (for example, at espn.go.com) you find the salaries of all 22 players for the Chicago Cubs as of opening day of the 2005 baseball season. The club total was $87 million, eighth in the major leagues. Which inference procedure would you use to estimate the average salary of the Cubs players? (a) one-sample z interval for (b) one-sample t interval for (c) one-sample t test (d) one-sample z test (e) none of these 8. You read in the report of a psychology experiment that “separate analyses for our two groups of 12 participants revealed no overall placebo effect for our student group (mean = 0.08, SD = 0.37, t(11) = 0.49) and a significant effect for our non-student group (mean = 0.35, SD = 0.37, t(11) = 3.28, p < 0.01).” Are the two values given for the t test statistic correct? (The null hypothesis is that the mean effect is zero.) (a) Yes, both are correct. (b) The t statistic for the student group is correct, but the one for the non-student group is incorrect. (c) The t statistic for the non-student group is correct, but the one for the student group is incorrect. (d) Both t statistics are incorrect. (e) We can’t tell whether either t statistic is correct, because we aren’t given the actual data. PART IV: 1. A sociologist is studying the effect of having children within the first two years of marriage on the divorce rate. Using hospital birth records, she selects a random sample of 200 couples who had a child within the first two years of marriage. Following up on these couples, she finds that 80 couples are divorced within five years. To determine if having children within the first two years of marriage increases the divorce rate we should test (a) hypotheses H0: p = 0.50, Ha: p 0.50. (b) hypotheses H0: p = 0.50, Ha: p > 0.50. (c) hypotheses H0: p = 0.50, Ha: p < 0.50. (d) hypotheses H0: p = 0.40, Ha: p > 0.40. (e) none of the above. Chapter 11 5 Test 11A 2. In order to study the amounts owed to a particular city, a city clerk takes a random sample of 16 files from a cabinet containing a large number of delinquent accounts and finds the average amount x owed to the city to be $230 with a sample standard deviation of $36. It has been claimed that the true mean amount owed on accounts of this type is greater than $250. If it is appropriate to assume that the amount owed is a Normally distributed random variable, the value of the test statistic appropriate for testing the claim is (a) 3.33 (b) 1.96 (c) 2.22 (d) 0.55 (e) 2.1314 3. An inspector inspects large truckloads of potatoes to determine the proportion p in the shipment with major defects prior to using the potatoes to make potato chips. Unless there is clear evidence that this proportion is less than 0.10, she will reject the shipment. To reach a decision she will test the hypotheses H0: p = 0.10, Ha: p < 0.10 using the large-sample test for a population proportion. To do so, she selects an SRS of 50 potatoes from the more than 2000 potatoes on the truck. Suppose that only two of the potatoes sampled are found to have major defects. Which of the following conditions for inference about a proportion using a hypothesis test are violated? (a) The data are an SRS from the population of interest. (b) The population is at least 10 times as large as the sample. (c) n is so large that both np0 and n(1 – p0) are 10 or more, where p0 is the proportion with major defects if the null hypothesis is true. (d) There appear to be no violations. (e) More than one condition is violated. 4. What is the value of t * , the critical value of the t distribution with 8 degrees of freedom, which satisfies the condition that the probability is 0.10 of being larger than t * ? (a) 1.415 (b) 1.397 (c) 1.645 (d) 2.896 (e) 0.90 5. The water diet requires one to drink two cups of water every half hour from when one gets up until one goes to bed, but otherwise allows one to eat whatever one likes. Four adult volunteers agree to test the diet. They are weighed prior to beginning the diet and after six weeks on the diet. The weights (in pounds) are Person Weight before the diet Weight after six weeks 1 180 170 2 125 130 3 240 215 4__ 150 152 For the population of all adults, assume that the weight loss after six weeks on the diet (weight before beginning the diet – weight after six weeks on the diet) is Normally distributed with mean µ. To determine if the diet leads to weight loss, we test the hypotheses H0: = 0, Ha: > 0 Based on these data we conclude that (a) we would not reject H0 at significance level 0.10. (b) we would reject H0 at significance level 0.10 but not at 0.05. (c) we would reject H0 at significance level 0.05 but not at 0.01. (d) we would reject H0 at significance level 0.01. (e) the sample size is too small to allow use of the t procedures. 6. Because t procedures are robust, the most important condition for their use is (a) the population standard deviation is known (b) the population distribution is exactly Normal (c) the data can be regarded as an SRS from the population (d) np and n(1 – p) are both at least 10 (e) there are no outliers in the sample data Chapter 11 6 Test 11A 7. Which of the following 95% confidence intervals would lead us to reject H 0 : p 0.30 in favor of H a : p 0.30 at the 5% significance level? (a) (0.30, 0.38) (b) (0.19, 0.27) (c) (0.27, 0.31) (d) (0.24, 0.30) (e) None of these 8. A medical researcher wishes to investigate the effectiveness of exercise versus diet in losing weight. Two groups of 25 overweight adult subjects are used, with a subject in each group matched to a similar subject in the other group on the basis of a number of physiological variables. One of the groups is placed on a regular program of vigorous exercise but with no restriction on diet, and the other is placed on a strict diet but with no requirement to exercise. The weight losses after 20 weeks are determined for each subject, and the difference between matched pairs of subjects (weight loss of subject in exercise group weight loss of matched subject in diet group) is computed. The mean of these differences in weight loss is found to be 2 lb with standard deviation s = 4 lb. Is this evidence of a difference in mean weight loss for the two methods? To answer this question, you should use (a) one-proportion z test (b) one-sample t test (c) one-sample z test (d) one-proportion z interval (e) one-sample t interval Chapter 11 7 Test 11A Chapter 11 8 Test 11A