# Chapter Review

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1 Chapter Review Name 1. In formulating hypotheses for a statistical test of significance, the null hypothesis is often a statement of no effect or no difference. the probability of observing the data you actually obtained. a statement that the data are all the probability that the parameter value is actually. 2. In their advertisements, a new diet program would like to claim that their methods result in a mean weight loss of more than ten pounds in two weeks. In order to determine if this is a valid claim, they hire an independent testing agency that then selects twenty-five people to be placed on this diet. The agency should be testing the null hypothesis H0: = 10 and the alternative hypothesis 3. The P-value of a test of a null hypothesis is the probability that assuming the null hypothesis is true, the test statistic will take a value at least as extreme as that actually observed. assuming the null hypothesis is false, the test statistic will take a value at least as extreme as that actually observed. the null hypothesis is true. the null hypothesis is false. the alternative hypothesis is true. 4. To determine if having children within the first two years of marriage increases the divorce rate, where p = proportion of marriages that end in divorce, we should test the hypotheses 5. One effect of the pesticide 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. 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 6. In testing hypotheses, which of the following would be strong evidence against the null hypothesis? Using a small level of significance Using a large level of significance Obtaining data with a small P-value Obtaining data with a large P-value Obtaining data with a low test statistic 7. In tests of significance about an unknown parameter of some population, which of the following is considered strong evidence against the null hypothesis? The value of an estimate of the unknown parameter based on a simple random sample from the population is not equal to zero. The value of an estimate of the unknown parameter lies within 2 units of the sample value. We observe a value of an estimate of the unknown parameter based on a simple random sample from the population that is very consistent with the null hypothesis. We observe a value of an estimate of the unknown parameter based on a simple random sample from the population that is very unlikely to occur if the null hypothesis is true. The value of an estimate of the unknown parameter based on a simple random sample from the population is equal to zero. Chapter Review

2 8. In a test of statistical hypotheses, the P-value tells us if the null hypothesis is true. if the alternative hypothesis is true. the largest level of significance at which the null hypothesis can be rejected. the smallest level of significance at which the null hypothesis can be rejected. the probability of obtaining a sample with a test statistic that is farther from 0 than the one we obtained. 9. A test of significance produces a P-value of Which of the following conclusions is appropriate? Accept H a at the = 0.05 level Reject H a at the = 0.01 level Fail to reject H 0 at the =.05 level Reject H 0 at the = 0.05 level Accept H 0 at the = 0.01 level 10. A university administrator obtains a sample of the academic records of past and present scholarship athletes at the university. The administrator reports that no significant difference was found in the mean GPA (grade point average) for male and female scholarship athletes (P = 0.287). This means that the GPAs for male and female scholarship athletes are identical, except for 28.7% of the athletes. the maximum difference in GPAs between male and female scholarship athletes is the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample, if there is no difference in mean GPAs, is the chance that a pair of randomly chosen male and female scholarship athletes would have a significant difference in GPAs is the probability that female athletes have higher GPAs than males do is I conduct a statistical test of hypotheses and find that the null hypothesis is statistically significant at level = I may conclude that the test would also be significant at level = the test would also be significant at level = the P-value is less than.05. both (A) and (C) are true. both (B) and (C) are true. 12. If we reject the null hypothesis when, in fact, it is true, we have committed a Type I error. committed a Type II error. a probability of being correct that is equal to the P-value. a probability of being correct that is equal to 1 P-value. set the level too high. 13. A Type II error is rejecting the null hypothesis when it is true. failing to reject the null hypothesis when it is false. incorrectly specifying the null hypothesis. incorrectly specifying the alternative hypothesis. more serious than a Type I error. 14. A researcher plans to conduct a test of hypotheses at the = 0.01 significance level. She designs her study to have a power of 0.90 at a particular alternative value of the parameter of interest. The probability that the researcher will commit a Type I error is equal to the P-value and cannot be determined until the data have been collected. 15. A researcher plans to conduct a test of hypotheses at the = 0.01 significance level. She designs her study to have a power of 0.90 at a particular alternative value of the parameter of interest. The probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power is equal to the 1 - (P-value) and cannot be determined until the data have been collected. Chapter Review

3 16. The power of a statistical test of hypotheses is the smallest significance level at which the data will allow you to reject the null hypothesis. equal to 1 - (P-value). the probability that the test will reject both one-sided and two-sided hypotheses. the probability that a significance test will reject the null hypothesis when a particular alternative value of the parameter is true. equal to 1 - P (Type I error). 17. Which of the following will increase the power of a statistical test of significance. Increase the Type II error probability. Increase the sample size. Reject the null hypothesis only if the P-value is smaller than the level of significance. Decrease the level. All of the above. 18. In hypothesis testing, is the probability of committing a Type II error. The power of the test,, is then the probability of rejecting H 0 when H a is true the probability of failing to reject H 0 when H a is true the probability of failing to reject H 0 when H 0 is true the probability of rejecting H 0 when H 0 is true the probability of failing to reject H An advertiser wishes to see if a new advertisement is effective in promoting an existing product. The previous advertisement has a recognition score of 3.7. An SRS of 12 potential buyers resulted in a mean recognition score of 3.4 with a standard deviation of 1.7. Which of the following required conditions for conducting a t-test for a mean has not been met? The population is at least 10 times the sample size. The data are taken from a simple random sample. The population is Normally distributed or n is large. The decision of each buyer is independent. All of the required conditions are met. Scenario 9-3 A December 2007 Gallup Poll reported that 43% of Americans use the internet for an hour or more each day. You suspect that a higher proportion of students at your school use the internet that much. To find out, you take a simple random sample of 60 students and find that 35 of them use the internet for an hour or more each day. (Assume your school has enough students so that this is a small sample relative to the size of the population). You will test the hypotheses, where p = the proportion of students at your school who use the internet for an hour or more each day, at the = 0.01 level. 20. Use Scenario 9-3. The test statistic, P-value, and appropriate decision for this test are: z = 2.40; P-value = 0.008; reject H o z = 2.40; P-value = 0.008; fail to reject H o t = 2.40; P-value = ; reject H o t = 2.40; P-value = ; fail to reject H o no conclusion can be drawn, because the shape of the sampling distribution is unknown. 21. In 1999, 2.2% of all cars in the United States were reported stolen. In a random sample of 400 Nissan Maxima cars that year, 12 were reported stolen. Is this evidence (at the = 0.05 level) that the theft rate for this model is higher than the national rate? Yes, the P-value = , so we reject H 0 and conclude that the rate for Nissans is higher that 2.2%. No, the P-value = , so we fail to reject H 0 and cannot conclude that the rate for Nissans is higher that 2.2%. Yes, the P-value = , so we reject H 0 and conclude that the rate for Nissans is higher that 2.2%. No, the P-value = , so we fail to reject H 0 and cannot conclude that the rate for Nissans is higher that 2.2%. We cannot perform this test because the conditions for inference have not been met. Chapter Review

4 Scenario 9-4 An SRS of 100 postal employees found that the average time these employees had worked for the postal service was J = 7 years with standard deviation s = 2 years. Assume the distribution of the time the population of employees have worked for the postal service is approximately Normal with mean. Are these data evidence that has changed from the value of 7.5 years of 20 years ago? To make this determination we test the hypotheses H 0 : = 7.5, H a : 7.5 using a one-sample t test. 22. Use Scenario 9-4. Which of the following intervals contains the P-value for this test? larger than 0.10 between 0.10 and 0.05 between 0.05 and 0.01 below 0.01 The P-value cannot be determined, since the standard deviation of the population from 20 years ago is not known. 23. Use Scenario 9-4. Suppose the mean and standard deviation we obtained were based on a sample of 25 postal workers, rather than 100. The P-value would be larger. smaller. unchanged, since the difference between and the hypothesized value = 7.5 is unchanged. unchanged, since both groups of workers have the same type of job. unchanged, since the variability measured by the standard deviation stays the same. 24. You construct a 95% confidence interval for a mean and find it to be 1.1 ± 0.8. Which of the following is true? A test of the hypotheses H 0 : = 1.2, H a : 1.2 would reject H 0 at the 0.05 level. A test of the hypotheses H 0 : = 1.1, H a : 1.1 would reject H 0 at the 0.05 level. A test of the hypotheses H 0 : = 0, H a : 0 would reject H 0 at the 0.05 level. All three tests above would reject H 0 at the 0.05 level. A test of hypothesis cannot be performed from only a confidence interval. 25. I wish to test the hypothesis based on an SRS of size n from a Normal population. I calculate a 95% confidence interval for and find it to be 1.33 to Which of the following is true? I would reject H 0 at level.05. I would reject H a at level.05. The P-value is.05. A mistake has almost certainly been made. The confidence interval must contain = 5 at least 95% of the time. None of these. 26. You test the hypothesis against the alternative and obtain a P-value of Which of the following must be true? A 95% confidence interval for will include the value 1. A 95% confidence interval for will include the value 0. A 99% confidence interval for will include the value 1. A 99% confidence interval for will include the value 0. None of these is necessarily true. 27. You have data on rainwater collected at 16 locations in the Adirondack Mountains of New York State. One measurement is the acidity of the water, measured by ph on a scale of 0 to 14 (the ph of distilled water is 7.0). Which inference procedure would you use to estimate the average acidity of rainwater in the Adirondacks? one-sample z interval for one-sample t interval for one proportion z-test one-sample t test one-sample z test Chapter Review

5 Chapter Review Answer Section MULTIPLE CHOICE 1. ANS: A PTS: 1 TOP: Idea of null hypothesis 2. ANS: B PTS: 1 TOP: Stating hypotheses 3. ANS: A PTS: 1 TOP: Idea of P-value 4. ANS: E PTS: 1 TOP: Stating hypotheses 5. ANS: A PTS: 1 TOP: Stating hypotheses 6. ANS: C PTS: 1 TOP: Idea of P-value 7. ANS: D PTS: 1 TOP: Idea of P-value 8. ANS: E PTS: 1 TOP: Idea of P-value 9. ANS: D PTS: 1 TOP: P-value only gives evidence against null 10. ANS: C PTS: 1 TOP: Idea of P-value 11. ANS: D PTS: 1 TOP: Statistical significance 12. ANS: A PTS: 1 TOP: Type I and II error 13. ANS: B PTS: 1 TOP: Type I and II error 14. ANS: A PTS: 1 TOP: Type I and II error 15. ANS: B PTS: 1 TOP: Type I and II error 16. ANS: D PTS: 1 TOP: Power of a test 17. ANS: B PTS: 1 TOP: Power of a test 18. ANS: A PTS: 1 TOP: Type I and II error 19. ANS: C PTS: 1 TOP: Conditions for test of proportions 20. ANS: A PTS: 1 TOP: Calculations for proportions test 21. ANS: E PTS: 1 TOP: Conditions for test of proportions 22. ANS: C PTS: 1 TOP: T-test: find P-value 23. ANS: A PTS: 1 TOP: T-test: impact of non P-value 24. ANS: C PTS: 1 TOP: Link conf. interval and test of signif. 25. ANS: A PTS: 1 TOP: Link conf. interval and test of signif. 26. ANS: C PTS: 1 TOP: Link conf. interval and test of signif. 27. ANS: B PTS: 1 TOP: Choosing a procedure Chapter Review

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