CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING

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1 CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized test statistic z equals: a b c d C 57. If a hypothesis is not rejected at the 0.10 level of significance, it: a. must be rejected at the 0.05 level. b. may be rejected at the 0.05 level. c. will not be rejected at the 0.05 level. d. must be rejected at the level. C 58. In order to determine the p-value, which of the following is not needed? a. The level of significance. b. Whether the test is one-tail or two-tail. c. The value of the test statistic. d. All of these choices are true. A 59. In testing the hypotheses H 0 : µ = 75 vs. H 1 : µ < 75, if the value of the test statistic z equals 2.42, then the p-value is: a b c d D 60. For a two-tail test, the null hypothesis will be rejected at the 0.05 level of significance if the value of the standardized test statistic z is: a. smaller than 1.96 or greater than 1.96 b. greater than 1.96 or smaller than 1.96 c. smaller than 1.96 or greater than 1.96 d. greater than or less than C

2 61. In testing the hypotheses H 0 : µ = 800 vs. H 1 : µ 800, if the value of the test statistic equals 1.75, then the p-value is: a b c d B 62. If a hypothesis is rejected at the level of significance, it: a. must be rejected at any level. b. must be rejected at the 0.01 level. c. must not be rejected at the 0.01 level. d. may or may not be rejected at the 0.01 level. D 63. Which of the following p-values will lead us to reject the null hypothesis if the level of significance equals 0.05? a b c d D 64. Suppose that we reject a null hypothesis at the 0.05 level of significance. Then for which of the following α-values do we also reject the null hypothesis? a b c d A 65. The critical values z α or z α / 2 are the boundary values for: a. the rejection region(s). b. the level of significance. c. Type I error. d. Type II error. A 66. In a two-tail test for the population mean, if the null hypothesis is rejected when the alternative hypothesis is true: a. a Type I error is committed. b. a Type II error is committed. c. a correct decision is made. d. a one-tail test should be used instead of a two-tail test. C

3 67. Using a confidence interval when conducting a two-tail test for µ, we do not reject H 0 if the hypothesized value for µ: a. is to the left of the lower confidence limit (LCL). b. is to the right of the upper confidence limit (UCL). c. falls between the LCL and UCL. d. falls in the rejection region. C 68. In a two-tail test for the population mean, the null hypothesis will be rejected at α level of significance if the value of the standardized test statistic z is such that: a. z > z α b. z < z α c. z α < z < z α d. z > z α / 2 D 69. In testing the hypothesis H 0 : µ = 100 vs. H 1 : µ > 100, the p-value is found to be 0.074, and the sample mean is 105. Which of the following statements is true? a. The probability of observing a sample mean at least as large as 105 from a population whose mean is 100 is b. The probability of observing a sample mean smaller than 105 from a population whose mean is 100 is c. The probability that the population mean is larger than 100 is d. None of these choices. A 70. If we reject the null hypothesis, we conclude that: a. there is enough statistical evidence to infer that the alternative hypothesis is true. b. there is not enough statistical evidence to infer that the alternative hypothesis is true. c. there is enough statistical evidence to infer that the null hypothesis is true. d. there is not enough statistical evidence to infer that the null hypothesis is true. A 71. Suppose that in a certain hypothesis test the null hypothesis is rejected at the.10 level; it is also rejected at the.05 level; however it cannot be rejected at the.01 level. The most accurate statement that can be made about the p-value for this test is that: a. p-value = b. p-value = c < p-value < d < p-value < C

4 72. Statisticians can translate p-values into several descriptive terms. Suppose you typically reject H 0 at level Which of the following statements is correct? a. If the p-value < 0.001, there is overwhelming evidence to infer that the alternative hypothesis is true. b. If 0.01 < p-value < 0.05, there is evidence to infer that the alternative hypothesis is true. c. If p-value > 0.10, there is no evidence to infer that the alternative hypothesis is true. d. All of these choices are true. D 73. If we do not reject the null hypothesis, we conclude that: a. there is enough statistical evidence to infer that the alternative hypothesis is true. b. there is not enough statistical evidence to infer that the alternative hypothesis is true. c. there is enough statistical evidence to infer that the null hypothesis is true. d. there is not enough statistical evidence to infer that the null hypothesis is true. B 74. In a one-tail test, the p-value is found to be equal to If the test had been two-tail, the p-value would have been: a b c d D 75. If the value of the sample mean is close enough to the hypothesized value µ 0 of the population mean µ, then: a. the value of µ 0 is definitely correct. b. the value of µ 0 is definitely wrong. c. we reject the null hypothesis. d. we cannot reject the null hypothesis. D 76. The p-value of a test is the: a. smallest α at which the null hypothesis can be rejected. b. largest α at which the null hypothesis can be rejected. c. smallest α at which the null hypothesis cannot be rejected. d. largest α at which the null hypothesis cannot be rejected. A 77. We have created a 95% confidence interval for µ with the result (8, 13). What conclusion will we make if we test H 0 : µ = 15 vs. H 1 : µ 15 at α = 0.05? a. Reject H 0 in favor of H 1 b. Accept H 0 in favor of H 1 c. Fail to reject H 0 in favor of H 1 d. We cannot tell what our decision will be from the information given A

5 78. The p-value criterion for hypothesis testing is to reject the null hypothesis if: a. p-value = α b. p-value < α c. p-value > α d. α < p-value < α B 79. If the p value is less than α in a two-tail test: a. the null hypothesis should not be rejected. b. the null hypothesis should be rejected. c. a one-tail test should be used. d. No conclusion should be reached. B 80. If an economist wishes to determine whether there is evidence that average family income in a community exceeds $32,000: a. either a one-tail or two-tail test could be used with equivalent results. b. a one-tail test should be utilized. c. a two-tail test should be utilized. d. None of these choices. B 81. We have created a 95% confidence interval for µ with the results (10, 25). What conclusion will we make if we test H 0 : µ = 26 vs. H 1 : µ 26 at α = 0.025? a. Reject H 0 in favor of H 1 b. Accept H 0 in favor of H 1 c. Fail to reject H 0 in favor of H 1 d. We cannot tell from the information given. D 82. The rejection region for testing H 0 : µ = 100 vs. H 1 : µ 100, at the 0.05 level of significance is: a. z < 0.95 b. z > 1.96 c. z > 1.65 d. z < 2.33 B 83. The owner of a local nightclub has recently surveyed a random sample of n = 300 customers of the club. She would now like to determine whether or not the mean age of her customers is over 35. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. Suppose she found that the sample mean was 35.5 years and the population standard deviation was 5 years. What is the p-value associated with the test statistic? a b c d

6 C 84. It is possible to directly compare the results of a confidence interval estimate to the results obtained by testing a null hypothesis if: a. a two-tail test for µ is used. b. a one-tail test for µ is used. c. a two-tail test for is used. d. a one-tail test for is used. A 85. The rejection region for testing H 0 : µ = 80 vs. H 1 : µ < 80, at the 0.10 level of significance is: a. z > 1.96 b. z < 0.90 c. z > 1.28 d. z < 1.28 D 86. We have created a 90% confidence interval for µ with the result (25, 32). What conclusion will we make if we test H 0 : µ = 28 vs. H 1 : µ 28 at α = 0.10? a. Reject H 0 in favor of H 1. b. Accept H 0 in favor of H 1. c. Fail to reject H 0 in favor of H 1. d. We cannot tell from the information given. C 87. The numerical quantity computed from the data that is used in deciding whether to reject H 0 is the: a. significance level. b. critical value. c. test statistic. d. parameter. C 88. The owner of a local nightclub has recently surveyed a random sample of n = 300 customers of the club. She would now like to determine whether or not the mean age of her customers is over 35. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. If she wants to be 99% confident in her decision, what rejection region she use if the population standard deviation σ is known? a. Reject H 0 if z < 2.33 b. Reject H 0 if z < 2.58 c. Reject H 0 if z > 2.33 d. Reject H 0 if z > 2.58 C TRUE/FALSE 89. A one-tail p-value is two times the size of a two-tail test. F

7 90. A p-value is usually set at F 91. The p-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true. T 92. For a given level of significance, if the sample size is increased, the probability of committing a Type II error will decrease. T 93. The critical values will bound the rejection and non-rejection regions for the null hypothesis. T 94. If we do not reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the null hypothesis is true. F 95. The p-value of a test is the smallest α at which the null hypothesis can be rejected. T 96. If a null hypothesis is rejected at the 0.05 level of significance, it must be rejected at the level. F 97. In a one-tail test, the p-value is found to be equal to If the test had been two-tail, then the p- value would have been F 98. A sample is used to obtain a 95% confidence interval for the mean of a population. The confidence interval goes from to If the same sample had been used to test the null hypothesis that the mean of the population differs from 90, the null hypothesis could be rejected at a level of significance of T 99. The p-value is the probability that the null hypothesis is true. F 100. In order to determine the p-value, it is necessary to know the level of significance. F

8 101. If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10 level. T 102. If your p-value is greater than you should reject H 0 at the 0.10 level. F 103. A p-value is a probability, and must be between 0 and 1. T 104. A one-tail test for the population mean µ produces a test-statistic z = The p-value associated with the test is F 105. Using the confidence interval when conducting a two-tail test for the population mean µ, we do not reject the null hypothesis if the hypothesized value for µ falls between the lower and upper confidence limits. T 106. A two-tail test for the population mean µ produces a test-statistic z = The p-value associated with the test is T 107. For a given level of significance, if the sample size is increased, the probability of committing a Type I error will decrease. F 108. The larger the p-value, the more likely one is to reject the null hypothesis. F 109. A sample is used to obtain a 95% confidence interval for the mean of a population. The confidence interval goes from to If the same sample had been used to test H 0 : µ = 12 vs. H 1 : µ 12, H 0 could not be rejected at the 0.05 level. T 110. If we reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the alternative hypothesis is true. T

9 COMPLETION 111. There are two approaches to making a decision in a hypothesis test once the test statistic has been calculated. One approach is the method. The other approach is the p-value method. rejection region 112. There are two approaches to making a decision in a hypothesis test once the test statistic has been calculated. One approach is the rejection region method. The other approach is the method. p-value p value 113. The is a range of values such that if the test statistic falls into that range we reject the null hypothesis. rejection region 114. The probability of a test statistic falling in the rejection region is equal to the value of. α the significance level a Type I error 115. When a null hypothesis is rejected, the test is said to be statistically at level α. significant 116. If the conclusion of a hypothesis test is that a statistically significant result was found, then the null hypothesis (was/was not) rejected. was

10 117. The of a test is the probability of observing a test statistic at least as extreme as the one from your sample, given that H 0 is true. p-value p value 118. You reject H 0 if the p-value of your hypothesis is than the significance level. less smaller 119. The is a measure of the amount of statistical evidence that supports the alternative hypothesis. p-value p value 120. If we do not reject the null hypothesis, we conclude that there (is/is not) enough statistical evidence to infer that the alternative hypothesis is true. is not SHORT ANSWER Production Filling A production filling operation has a historical standard deviation of 6 ounces. When in proper adjustment, the mean filling weight for the production process is 50 ounces. A quality control inspector periodically selects at random 36 containers and uses the sample mean filling weight to see if the process is in proper adjustment {Production Filling Narrative} State the null and alternative hypotheses. H 0 : µ = 50 vs. H 1 : µ 50

11 122. {Production Filling Narrative} Using a standardized test statistic, test the hypothesis at the 5% level of significance if the sample mean filling weight is 48.6 ounces. Test statistic: z = 1.40 Rejection region: z > z.025 = 1.96 Conclusion: Don't reject H 0. We cannot infer that the process is out of proper adjustment {Production Filling Narrative} Develop a 95% confidence interval and use it to test the hypothesis. LCL = 46.64, and UCL = Since the hypothesized value 50 falls in the 95% confidence interval, we fail to reject H 0 at α = A social scientist claims that the average adult watches less than 26 hours of television per week. He collects data on 25 individuals' television viewing habits and finds that the mean number of hours that the 25 people spent watching television was 22.4 hours. If the population standard deviation is known to be eight hours, can we conclude at the 1% significance level that he is right? H 0 : µ = 26 vs. H 1 : µ < 26 Test statistic: z = 2.25 Rejection region: z < z.01 = 2.33 Conclusion: Don't reject H 0. No, we cannot conclude at α =.01 that the social scientist is right A random sample of 100 observations from a normal population whose standard deviation is 50 produced a mean of 75. Does this statistic provide sufficient evidence at the 5% level of significance to infer that the population mean is not 80? H 0 : µ = 80 vs. H 1 : µ 80 Rejection region: z > z.025 = 1.96 Test statistic: z = 1.0 Conclusion: Don't reject H 0. Insufficient evidence at the 5% level to say the population mean differs from 80.

12 126. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ < 50, we found that the standardized test statistic is z = Calculate the p-value, and state your conclusion if α =.025. p-value = > α. Fail to reject H 0. Insufficient evidence to say the population mean is less than Suppose that 10 observations are drawn from a normal population whose variance is 64. The observations are: 58, 62, 45, 50, 59, 65, 39, 40, 41, and 52. Test at the 10% level of significance to determine if there is enough evidence to conclude that the population mean is greater than 45. H 0 : µ = 45, H 1 : µ > 45 Test statistic: z = 2.41 p-value = Reject H 0. Yes, there is enough statistical evidence at the 10% significance level to conclude that the population mean is greater than Suppose that 9 observations are drawn from a normal population whose standard deviation is 2. The observations are: 15, 9, 13, 11, 8, 12, 11, 7, and 10. At 95% confidence, you want to determine whether the mean of the population from which this sample was taken is significantly different from 10. a. State the null and alternative hypotheses. b. Compute the value of the test statistic. c. Compute the p-value. d. Interpret the results. a. H 0 : µ = 10 vs. H 1 : µ 10 b. z = 1.0 c = d. Cannot reject H 0. Not enough evidence to say the mean is other than Determine the p-value associated with each of the following values of the standardized test statistic z, and state your conclusion. a. two-tail test, with z = 1.50, and α =.10 b. one-tail test, with z = 1.05, and α =.05 c. one-tail test, with z = 2.40, and α =.01 a , fail to reject H 0 b , fail to reject H 0 c , reject H 0

13 Watching Sports A researcher claims athletes spend an average of 40 minutes per day watching sports. You think the average is higher than that. In testing your hypotheses H 0 : µ = 40 vs. H 1 : µ > 40, the following information came from your random sample of athletes: = 42 minutes, n = 25. Assume σ = 5.5, and α = {Watching Sports Narrative} Calculate the value of the test statistic. z = {Watching Sports Narrative} Set up the rejection region. z > z.05 = {Watching Sports Narrative} Determine the p-value {Watching Sports Narrative} Interpret the result. Reject H 0 and conclude that the average time watching sports is more than 40 minutes per day for athletes (at the 0.10 level). GRE Scores The Admissions officer for the graduate programs at Michigan State University (MSU) believes that the average score on the GRE exam at his university is significantly higher than the national average of 1,300. An accepted standard deviation for GRE scores is 125. A random sample of 25 scores had an average of 1, {GRE Scores Narrative} State the appropriate null and alternative hypotheses. H 0 : µ = 1300 vs. H 1 : µ > 1300

14 135. {GRE Scores Narrative} Calculate the value of the test statistic and set up the rejection region at the level. What is your conclusion? Test statistic: z = 3.0 Rejection region: z > z.025 = 1.96 Conclusion: Reject H 0. There is enough statistical evidence to infer that the average GRE for all graduate students at MSU is higher than 1, {GRE Scores Narrative} Calculate the p-value. p-value = {GRE Scores Narrative} Use the p-value to test the hypotheses. Since p-value = < α = 0.025, we reject H With the following p-values, would you reject or fail to reject the null hypothesis? Comment on the statistical significance of each result. (Assume you normally reject H 0 at level 0.08.) a. p-value = b. p-value = c. p-value = d. p-value = a. There is overwhelming evidence to reject H 0. The test is highly significant. b. There is strong evidence to reject H 0. The test is deemed to be significant. c. There is weak evidence to reject H 0. The test is marginally statistically significant. d. There is no evidence to reject H 0. The test is not statistically significant. Runners A researcher wants to study the average miles run per day for runners. In testing the hypotheses: H 0 : µ = 25 miles vs. H 1 : µ 25 miles, a random sample of 36 runners drawn from a normal population whose standard deviation is 10, produced a mean of 22.8 miles weekly {Runners Narrative} Compute the value of the test statistic and specify the rejection region associated with 5% significance level.

15 Test statistic: z = 1.32 Rejection region: z > z.025 = {Runners Narrative} Compute the p-value = {Runners Narrative} What can we conclude at the 5% significance level regarding the null hypothesis? Fail to reject H 0. Not enough evidence to say the average runner mileage is different from {Runners Narrative} Develop a 95% confidence interval estimate of the population mean to {Runners Narrative} Explain briefly how to use the confidence interval to test the hypothesis. Since the hypothesized value µ 0 = 25 is included in the 95% confidence interval, we fail to reject H 0 at α = Microwave Oven An appliance manufacturer claims to have developed a new microwave oven that consumes an average of no more than 250 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a standard deviation of 18 W. A consumer group suspects the actual average is more than 250 W. They take a sample of 20 microwave ovens and calculate the average consumption to be 260 W {Microwave Oven Narrative} What is the parameter of interest in this situation? The mean power consumption of all such microwave ovens (in W).

16 145. {Microwave Oven Narrative} State the appropriate hypotheses for the consumer group to do their test. H 0 : µ = 250 vs. H 1 : µ > {Microwave Oven Narrative} For a test with a level of significance of 0.05, determine the critical value. z.05 = {Microwave Oven Narrative} What is the value of the test statistic? z = {Microwave Oven Narrative} Calculate the p-value of the test {Microwave Oven Narrative} What is the conclusion from the hypothesis test using α =.05? Since the p-value = <.05, we reject the null hypothesis. We conclude that the manufacturer's claim is false; that is, the new microwave oven consumes an average of more than 250 W. Due to the small size of the p-value, this result is highly statistically significant.

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