Unit 1

Unit 1 1. We take a simple random sample of 16 adults and ask them how long they sleep on a typical night. The sample mean is calculated to be 7.2 hours. Suppose it is known that the number of hours adults sleep at night follows a normal distribution with standard deviation 1.8 hours. An 88% confidence interval for the true mean time adults sleep at night is: (A) (6.80, 7.60) (B) (6.30, 8.10) (C) (6.60, 7.80) (D) (6.70, 7.70) (E) (6.50, 7.90) 2. The average daily calcium intake for a sample of 10 Canadian adults is calculated to be 615 mg. Daily calcium intake is known to follow a normal distribution with standard deviation 112 mg. An 83% confidence interval for the true mean daily calcium intake of all Canadian adults is: (A) (566.48, 663.52) (B) (564.45, 665.55) (C) (568.17, 661.83) (D) (565.73, 664.27) (E) (567.84, 662.16) 3. A random variable X follows a normal distribution with standard deviation σ = 8. We take a random sample of 25 individuals from the population and we calculate a confidence interval for μ to be (36.7136, 43.2864) . What is the confidence level of this interval? (A) 90% (B) 95% (C) 96% (D) 98% (E) 99%

4. We select a random sample of ten oranges from trees grown in a large orchard. The standard deviation of weights of these ten oranges is calculated to be 19 grams. Based on this sample, a confidence interval for the true mean weight of all oranges grown in the orchard is calculated to be (126 . 41 , 153 . 59). What is the confidence level of this interval? (A) 80% (B) 90% (C) 95% (D) 96% (E) 98% 5. The Manitoba Department of Agriculture would like to estimate the average yield per acre of a new variety of corn for farms in southwestern Manitoba. It is desired that the final estimate be within 5 bushels per acre of the true mean yield. Due to cost restraints, a sample of no more than 65 one-acre plots of land can be obtained to conduct an experiment. The population standard deviation is known to be 17.33 bushels per acre. What is (approximately) the maximum confidence level that could be attained for a confidence interval that meets the Agriculture Department’s specification? (A) 90% (B) 95% (C) 96% (D) 98% (E) 99% 6. We would like to construct a confidence interval to estimate the true mean systolic blood pressure of all healthy adults to within 3 mm Hg (millimetres of mercury). We have a sample of 36 adults available for testing. Systolic blood pressures of healthy adults are known to follow a normal distribution with standard deviation 14.04 mm Hg. What is the maximum confidence level that can be attained for our interval? (A) 80% (B) 90% (C) 95% (D) 96% (E) 98% 7. The sizes of farms in a U.S. state follow a normal distribution with standard deviation 30 acres. Suppose we measure the sizes of a random sample of farms and calculate a 98% confidence interval for μ to be (295, 305). What is the correct interpretation of this interval? (A) Approximately 98% of farms have a size between 295 and 305 acres. (B) Approximately 98% of samples of 30 farms will have a mean size between 295 and 305 acres. (C) The probability that the population mean is between 295 and 305 acres is 98%. (D) In repeated samples of the same size, 98% of similarly constructed intervals will contain the sample mean. (E) In repeated samples of the same size, 98% of similarly constructed intervals will contain the population mean. 2

12. The durations of professional men’s tennis matches are known to follow a normal dis- tribution. It is calculated that, in order to estimate the true mean duration to within 3 minutes with 95% confidence, a sample of 77 matches is required. What sample size is required in order to estimate the true mean duration to within 2 minutes with 95% confidence? (A) 35 (B) 52 (C) 95 (D) 116 (E) 174 13. We would like to estimate the true mean amount (in $) consumers spent last year on Christmas gifts. We record the amount spent for a simple random sample of 30 consumers and we calculate a 95% confidence interval for μ to be (500, 545), i.e., the length of the interval is 45. The standard deviation σ of the amount spent by consumers is known. Suppose we had instead selected a simple random sample of 90 consumers and calculated a 95% confidence interval for μ . What would be the length of this interval? (A) 5.00 (B) 12.99 (C) 15.00 (D) 25.98 (E) 77.94 14. A statistician conducted a test of H 0 : μ = 100 vs. H a : μ 6 = 100 for the mean μ of some normally distributed population. Based on the gathered data, the statistician calculated a sample mean of ¯ x = 110 and concluded that H 0 could be rejected at the 5% level of significance. Using the same data, which of the following statements must be true? (I) A test of H 0 : μ = 100 vs. H a : μ > 100 at the 5% level of significance would also lead to rejecting H 0 . (II) A test of H 0 : μ = 90 vs. H a : μ 6 = 90 at the 5% level of significance would also lead to rejecting H 0 . (III) A test of H 0 : μ = 100 vs. H a : μ 6 = 100 at the 1% level of significance would also lead to rejecting H 0 . (A) I only (B) II only (C) I and II (D) I and III (E) I, II and III 4

15. A statistician conducted a test of H 0 : μ = 1 vs. H a : μ > 1 for the mean μ of some population. Based on the gathered data, the statistician concluded that H 0 could be rejected at the 1% level of significance. Using the same data, which of the following statements must be true? (I) A test of H 0 : μ = 1 vs. H a : μ > 1 at the 10% level of significance would also lead to rejecting H 0 . (II) A test of H 0 : μ = 0 vs. H a : μ > 0 at the 1% level of significance would also lead to rejecting H 0 . (III) A test of H 0 : μ = 1 vs. H a : μ 6 = 1 at the 1% level of significance would also lead to rejecting H 0 . (A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II and III 16. We would like to conduct a hypothesis test to examine whether there is evidence that the true mean amount spent on textbooks by a U of M student in one semester differs from $400. A random sample of 50 students is selected and the mean amount they spent on textbooks for one semester is calculated to be $430. Assume the population standard deviation is known to be $165. What is the P-value for the appropriate hypothesis test? (A) 0.1970 (B) 0.1112 (C) 0.0630 (D) 0.0985 (E) 0.1260 17. A candy manufacturer sells boxes of its candy with an advertised weight of 50 grams. Weights of candy per box are known to follow a normal distribution with standard deviation 2.2 grams. A random sample of five boxes has a mean weight of 51.8 grams. We conduct a hypothesis test to determine if there is evidence that the true mean weight of all boxes of candy differs from the advertised weight. The P-value for the appropriate test of significance is: (A) 0.0336 (B) 0.0427 (C) 0.0548 (D) 0.0672 (E) impossible to determine, since the level of significance is not given. 5

21. The mean height of a random sample of 25 Major League Baseball (MLB) players is calculated to be 183 cm. We conduct a hypothesis test of H 0 : μ = 180 vs. H a : μ > 180 to determine whether the true mean height of all MLB players is greater than 180 cm. The P-value of the test is calculated to be 0.15. What is the correct interpretation of this P-value? (A) If the true mean height of all MLB players was 180 cm, the probability of rejecting the null hypothesis would be 0.15. (B) If we took repeated samples of 25 MLB players and conducted the test in a similar manner, 15% of all samples would have means at least as high as 180 cm. (C) The probability that the true mean height of all MLB players is greater than 180 cm is 0.15. (D) If the true mean height of all MLB players was 180 cm, the probability of observing a sample mean at least as high as 183 cm would be 0.15. (E) Given that the sample mean height is 183, the probability that the true mean height of all MLB players is greater than 180 cm is 0.15. 22. A test of H 0 : μ = 100 vs. H a : μ 6 = 100 is conducted for the mean μ of some population. We take a sample of ten individuals and calculate a sample mean of 104. A 98% confi- dence interval for μ is calculated to be (101, 107). Which of the following statements is true ? (A) We fail to reject H 0 at a 4% level of significance, since 100 is not contained in the 98% confidence interval. (B) We fail to reject H 0 at a 2% level of significance, since 104 is contained within the 98% confidence interval. (C) We reject H 0 at the 1% level of significance, since 100 is not contained within the 98% confidence interval. (D) We fail to reject H 0 at the 4% level of significance, since 104 is contained within the 98% confidence interval. (E) We reject H 0 at a 2% level of significance, since 100 is not contained within the 98% confidence interval. 7

23. We would like to conduct a hypothesis test to determine whether the true mean rent amount for all one-bedroom apartments in Winnipeg differs from $850. We take a random sample of 50 one-bedroom apartments and calculate the sample mean to be $900. A 98% confidence interval for μ is calculated to be (830, 970) . The conclusion for our test would be to: (A) fail to reject H 0 at the 2% level of significance since the value 850 is contained in the 98% confidence interval. (B) fail to reject H 0 at the 1% level of significance since the value 900 is contained in the 98% confidence interval. (C) reject H 0 at the 1% level of significance since the value 850 is contained in the 98% confidence interval. (D) reject H 0 at the 2% level of significance since the value 850 is contained in the 98% confidence interval. (E) fail to reject H 0 at the 4% level of significance since the value 900 is contained in the 98% confidence interval. 24. For which of the following tests would a Type II Error have more serious consequences than a Type I Error? (I) You are deciding whether to go for a skate on the frozen river. H 0 : The ice will not break. H a : The ice will break. (II) You are deciding whether to go for a swim in the ocean. H 0 : There are no sharks in the water. H a : There are sharks in the water. (III) You are wearing an expensive suit and you have to walk to work. You are deciding whether to take an umbrella with you. H 0 : It will not rain. H a : It will rain. (A) II only (B) I and II only (C) I and III only (D) II and III only (E) I, II and III 8

27. A farmer will measure the yields of a random sample of 15 of his tomato plants and conduct a hypothesis test to determine whether the true mean yield of all of his tomato plants is less than 20 pounds. Yields per plant are known to follow a normal distribution with standard deviation 4 pounds. The farmer decides to reject the null hypothesis if ¯ X ≤ 18 . 3 pounds. What is the probability of making a Type II error if the true mean yield of all of the farmer’s tomato plants is actually 17 pounds? (A) 0.0553 (B) 0.1038 (C) 0.1469 (D) 0.1736 (E) 0.2090 28. A hypothesis test of H 0 : μ = 75 vs. H a : μ > 75 is conducted at the 2% level of significance for some normally distributed variable X . It is calculated that, using a random sample of 25 individuals, if the true mean was in fact equal to 80, the power of the test would be equal to 0.90. All other things remaining the same , which of the following would result in a higher power? (I) taking a sample of 30 individuals (II) using a 1% level of significance (III) if the true mean was 85 (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III 29. In a hypothesis test, 1 - α is the probability of: (A) rejecting H 0 when H 0 is true. (B) rejecting H 0 when H a is true. (C) failing to reject H 0 when H 0 is true. (D) failing to reject H 0 when H a is true. (E) concluding that H 0 is true when H a is true. 10

30. The owner of a chemical fertilizer plant would like to conduct a hypothesis test at the 1% level of significance to determine whether the true mean amount of fertilizer produced at the plant is greater than 1000 pounds per day. He will record the amount of fertilizer produced at the plant for a random sample of 28 days. Fertilizer production at the plant is known to follow a normal distribution with standard deviation 189 pounds per day. What is the power of the test if the true mean amount of fertilizer produced is actually 1100 pounds per day? (A) 0.3192 (B) 0.4737 (C) 0.5263 (D) 0.6808 (E) 0.7760 The next two questions ( 31 and 32 ) refer to the following: The systems department of the Texaco Oil Company runs a large number of simulation programs on their mainframe computer. A manufacturer of new simulation software claims their program will run the simulation faster than the current mean of 30 minutes. The new program will be run 20 times. A hypothesis test of H 0 : μ = 30 vs. H a : μ < 30 is to be conducted, and it is decided that H 0 will be rejected if ¯ x ≤ 28 . 3 minutes. Run times for the new program are known to follow a normal distribution with standard deviation 3.7 minutes. 31. The level of significance of the test is closest to: (A) 0.01 (B) 0.02 (C) 0.03 (D) 0.04 (E) 0.05 32. What is the probability of making a Type II error if the true mean run time for the new program is actually 27 minutes? (A) 0.0192 (B) 0.0244 (C) 0.0384 (D) 0.0475 (E) 0.0582 33. We would like to conduct a hypothesis test at the 5% level of significance to determine whether there is evidence that the true mean age of first-time brides in 2017 has increased since 1997, when the true mean age was 25.4 years. We will take a random sample of 21 first-time brides. Ages of first-time brides in 2017 are known to follow a normal distribution with standard deviation 2.8 years. What is the power of the test if the true mean age of first-time brides in 2017 is 27 years? (A) 0.7881 (B) 0.8340 (C) 0.8962 (D) 0.9474 (E) 0.9738 34. Bottles of a certain brand of apple juice are supposed to contain 300 ml of juice. There is some variation from bottle to bottle because of imprecisions in the filling machinery. 11

38. The Winnipeg Free Press reported that the average size of all Manitoba farms in 2012 was 1135 acres. We take a random sample of 100 Manitoba farms in 2017 and conduct a hypothesis test of H 0 : μ = 1135 vs. H a : μ > 1135 at the 5% level of significance to determine whether the true mean size of Manitoba farms has increased over the last five years. Suppose it is known that farm sizes in Manitoba follow a normal distribution with standard deviation 252 acres. What is the probability of making a Type II error if the true mean size of farms in 2017 is 1200 acres? (A) 0.0427 (B) 0.0721 (C) 0.1093 (D) 0.1446 (E) 0.1762 39. We would like to conduct a hypothesis test for the mean μ of some population. We are trying to decide whether to use a level of significance of 1%, 5% or 10%. If we know that making a Type I error would have much more serious consequences than making a Type II error, then we should use: (A) a 1% level of significance, which will result in a higher power. (B) a 1% level of significance, which will result in a lower power. (C) a 5% level of significance, which will result in a power of 0.95. (D) a 10% level of significance, which will result in a higher power. (E) a 10% level of significance, which will result in a lower power. 40. We will take a random sample of 30 vehicles of a certain make and model and measure the fuel efficiency in miles per gallon (mpg) of each of them. We will conduct a hypothesis test at the 10% level of significance to determine whether there is evidence that the true mean fuel efficiency of all cars of this make and model differs from 32 mpg. What is the probability of failing to reject H 0 if the true mean is in fact 32 mpg? (A) 0.10 (B) 0.95 (C) 0.05 (D) 0.90 (E) 0.20 13

41. We conduct a hypothesis test of H 0 : μ = 20 vs. H a : μ 6 = 20 for the mean μ of some population at the 5% level of significance. Using a sample of size 30, we determine that H 0 should be rejected if ¯ x < 17 or ¯ x > 23. The power of the test against the alternative H a : μ = 25 is calculated to be 0.8621. Which of the following statements is true ? (A) If we had instead used a sample size of 50, the probability of a Type II error would increase. (B) The power of the test against the alternative H a : μ = 26 is less than 0.8621. (C) The power of the test against the alternative H a : μ = 15 is 0.1379. (D) If we had instead used a 1% level of significance, the probability of a Type II error would decrease. (E) If we had instead used a 10% level of significance, the power of the test would increase. 42. A consumer advocate suspects the true mean weight per bag of a certain brand of potato chips differs from 50 grams, the amount stated on the label. He will take a random sample of 25 bags of chips and conduct a hypothesis test at the 5% level of significance to test his suspicion. Weights per bag are known to follow a normal distribution. What is the probability of making a Type I Error if the null hypothesis is true? (A) 0.025 (B) 0.050 (C) 0.100 (D) impossible to determine without the value of ¯ x (E) impossible to determine without the value of σ 14

was calculated to be 7.8. What is the standard deviation of these six blood pressure measurements? (A) 2.8 (B) 3.2 (C) 14.4 (D) 19.1 (E) 46.8 16

47. We take a random sample of 12 observations from a normally distributed population. These 12 observations have a mean of 71.3 and standard deviation of 5.9. A confidence interval for μ is calculated to be (67.335, 75.265) . The confidence level of this interval is closest to: (A) 90% (B) 95% (C) 96% (D) 98% (E) 99% 48. The Winnipeg Transit Commission claims that the average time taken by the Number 60 bus to travel from the University of Manitoba to downtown is 27 minutes. A student who often takes this route believes that the true mean time is greater than 27 minutes. The student records the times for a sample of five trips. These trips have a mean time of 30 minutes and a standard deviation of 4 minutes. Suppose it is known that trip times follow a normal distribution. The P-value for the appropriate test of significance to test the student’s suspicion is: (A) between 0.01 and 0.02. (B) between 0.02 and 0.025. (C) between 0.025 and 0.05. (D) between 0.05 and 0.10. (E) between 0.10 and 0.15. 49. The dean of the Faculty of Science at a large university would like to conduct a hypothesis test to determine if the true mean GPA of science students at the university is less than 3.25. A random sample of 22 science students has a mean GPA of 3.18 and a standard deviation of 0.47. The P-value for the appropriate test of significance is: (A) between 0.15 and 0.20. (B) between 0.20 and 0.25. (C) between 0.40 and 0.50. (D) between 0.75 and 0.80. (E) The P-value cannot be calculated, as the level of significance was not given. 17

The next two questions ( 52 and 53 ) refer to the following: An experimenter is concerned about radiation levels in her research laboratory. A room is only considered safe if the mean radiation level is 400 or less. A random sample of 12 radiation measurements is taken at different locations in the laboratory. These 12 measurements have a mean of 414 and a standard deviation of 17. Radiation levels in the lab are known to follow a normal distribution with standard deviation 22. We would like to conduct a hypothesis test at the 5% level of significance to determine whether there is evidence that the laboratory is unsafe. 52. The test statistic for the appropriate test of significance is: (A) t = - 2 . 20 (B) z = - 2 . 20 (C) z = - 2 . 85 (D) t = 2 . 85 (E) z = 2 . 20 53. What is the correct conclusion for the test? (A) Reject H 0 . There is sufficient evidence that the lab is safe. (B) Fail to reject H 0 . There is insufficient evidence that the lab is unsafe. (C) Reject H 0 . There is sufficient evidence that the lab is unsafe. (D) Fail to reject H 0 . There is insufficient evidence that the lab is safe. (E) Reject H 0 . There is insufficient evidence that the lab is unsafe. 54. We would like to conduct a hypothesis test at the 10% level of significance to determine whether the true mean score of all players in a bowling league differs from 150. The mean and standard deviation of the scores of 12 randomly selected players are calculated to be 162 and 17, respectively. Scores of all players in the league are known to follow a normal distribution. Using the critical value method, the decision rule is to reject H 0 if the value of the test statistic is: (A) less than - 1 . 796 or greater than 1.796. (B) less than - 2 . 445 or greater than 2.445. (C) less than - 1 . 282 or greater than 1.282. (D) less than - 1 . 363 or greater than 1.363. (E) less than - 1 . 645 or greater than 1.645. 19

55. We will measure the weekly coffee consumption of a random sample of 15 Canadians, and we will conduct a hypothesis test at the 5% level of significance to determine whether the population mean differs from 3.0 litres per week. Coffee consumption is known to follow a normal distribution with unknown standard deviation σ . Using the critical value approach, the decision rule is to reject H 0 if the test statistic is: (A) less than - 1 . 761 or greater than 1.761. (B) greater than 1.645. (C) less than - 2 . 145 or greater than 2 . 145. (D) greater than 1.761. (E) less than - 1 . 960 or greater than 1 . 960. 20