BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test of hypotheses? A) Increase the significance level α. B) Increase the sample size. C) Consider computing the power for a value of the alternative that is farther from the value of the parameter of interest under the null hypothesis. D) All of the above. 2. Does vigorous exercise affect concentration? In general, the time needed for people to complete a certain paper and pencil maze follows a normal distribution, with a mean of 30 seconds and a standard deviation of three seconds. You wish to see if the mean time µ is changed by vigorous exercise, so you have a random sample of nine employees of your company (who you assume are representative of people in general) exercise vigorously for 30 minutes and then complete the maze. You compute the average x of their times to complete the maze and will use this information to test the hypotheses H0: µ = 30, Ha: µ 30. at the 1% significance level. The power of your test at µ = 28 seconds is approximately A) less than 0.001. B) 0.0630. C) 0.2810. D) 0.4877. A researcher plans to conduct a test of hypotheses at the 1% significance level. She designs her study to have a power of 0.90 at a particular alternative value of the parameter of interest. 3. The probability that the researcher will commit a type I error is A) 0.01. B) 0.10. C) 0.90. D) equal to the P-value and cannot be determined until the data have been collected. Page 1
4. 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 A) 0.01. B) 0.10. C) 0.90. D) equal to the 1 (P-value) and cannot be determined until the data have been collected. 5. A certain population follows a normal distribution with mean µ and standard deviation σ = 2.5. You collect data and test the hypotheses H0: µ = 1, Ha: µ 1. You obtain a P-value of 0.022. Which of the following is true? A) A 95% confidence interval for µ will include the value 1. B) A 95% confidence interval for µ will include the value 0. C) A 99% confidence interval for µ will include the value 1. D) A 99% confidence interval for µ will include the value 0. 6. A medical researcher is working on a new treatment for a certain type of cancer. The average survival time after diagnosis on the standard treatment is two years. In an early trial, she tries the new treatment on three subjects, who have an average survival time after diagnosis of four years. Although the survival time has doubled, the results are not statistically significant even at the 0.10 significance level. Suppose, in fact, that the new treatment does increase the mean survival time in the population of all patients with this particular type of cancer. The researcher has A) committed a type I error. B) committed a type II error. C) incorrectly used a level 0.10 test when she should have used a 0.05 level test. D) incorrectly used a level 0.10 test when she should have computed the P-value. 7. The power of a statistical test of hypotheses is A) the smallest significance level at which the data will allow you to reject the null hypothesis. B) equal to 1 (P-value). C) the extent to which the test will reject both one-sided and two-sided hypotheses. D) defined for a particular value of the parameter of interest under the alternative hypothesis and is the probability that a fixed level significance test will reject the null hypothesis when this particular alternative value of the parameter is true. Page 2
8. Ten years ago the mean Math SAT score of all high school students who took the exam in a small high school was 490, with a standard deviation of 80. This year, a boxplot of the scores of a random sample of 25 students in the high school who took the exam is given on the next page. The mean score of these 25 students is x = 530. We assume the population standard deviation continues to be σ = 80. To determine if there is evidence that the scores in the district have improved, the hypotheses H0: µ = 490, Ha: µ > 490 are tested using the z statistic and the P-value is found to be 0.0062. 825 750 675 600 525 450 We may conclude A) at the 5% significance level, you have proved that H0 is false. B) at the 5% significance level, you have proved that Ha is false. C) it would have been more appropriate to use a two-sided alternative. D) none of the above. Page 3
9. A researcher wishes to determine if listening to classical music improves the concentration of office workers. To investigate this, he decides to see if office workers are able to complete a certain pencil and paper maze more quickly while listening to classical music. Suppose the time (in seconds) needed for office workers to complete the maze while listening to classical music follows a normal distribution with mean µ and standard deviation σ = 4. Suppose also, that in the general population, the time needed to complete the maze (without listening to classical music) follows a normal distribution with mean 40 and standard deviation σ = 4. The researcher, therefore, decides to test the hypotheses H0: µ = 40, Ha: µ < 40. To do so, the researcher has 10,000 office workers complete the maze with classical music playing. The mean time for these workers is x = 39.8 seconds and the P-value is less than 0.0001. It is appropriate to conclude which of the following? A) The researcher has proved that for office workers, listening to classical music substantially improves the time it takes to complete the maze. B) The researcher has strong evidence that for office workers, listening to classical music substantially improves the time it takes to complete the maze. C) The researcher has moderate evidence that for office workers, listening to classical music substantially improves the time it takes to complete the maze. D) None of the above. 10. An engineer designs an improved light bulb. The previous design had an average lifetime of 1200 hours. The mean lifetime of a random sample of 2000 new bulbs is found to have a mean lifetime of 1201 hours. Although the difference from the old mean lifetime of 1200 hours is quite small, the P-value is 0.03 and the effect is statistically significant at the 0.05 level. If, in fact, there is no difference between the mean lifetimes of the new and old designs, the researcher has A) committed a type I error. B) committed a type II error. C) a probability of being correct, which is equal to the P-value. D) a probability of being correct, which is equal to 1 (P-value). 11. An agricultural researcher plants 25 plots with a new variety of corn that is drought resistant and hence potentially more profitable. The average yield for these plots is x = 150 bushels per acre. Assume that the yield per acre for the new variety of corn follows a normal distribution with unknown mean µ and that a 95% confidence interval for µ is found to be 150 ± 3.29. Which of the following is true? A) A test of the hypotheses H0: µ = 150, Ha: µ 150 would be rejected at the 0.05 level. B) A test of the hypotheses H0: µ = 150, Ha: µ > 150 would be rejected at the 0.05 level. C) A test of the hypotheses H0: µ = 160, Ha: µ 160 would be rejected at the 0.05 level. D) All the above. Page 4
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Answer Key 1. D 2. C 3. A 4. B 5. C Topic: 6.2 Tests of Significance 6. B 7. D 8. D Topic: 6.3 Use and Abuse of Tests 9. D Topic: 6.3 Use and Abuse of Tests 10. A 11. C Topic: 6.2 Tests of Significance Page 6