# MATHEMATICS

56) As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, = 19.8 and *S* = 5. Which of the following assumptions is necessary in order for a confidence interval to be valid?

A) The population sampled from has an approximate normal distribution.

B) The population sampled from has an approximate *t* distribution.

C) The mean of the sample equals the mean of the population.

D) None of these assumptions are necessary.

Answer: D

57) The county clerk wants to estimate the proportion of voters who will need special election facilities. Suppose a sample of 400 voters was taken. If 150 need special election facilities, calculate an 90% confidence interval for the population proportion.

Answer: 0.3352 to 0.4148

58) A quality control engineer is interested in estimating the proportion of defective items coming off a production line. In a sample of 300 items, 27 are defective. A 90% confidence interval for the proportion of defectives from this production line would go from ________ to ________.

Answer: 0.063 to 0.117

59) When determining the sample size for a proportion for a given level of confidence and sampling error, the closer to 0.50 that π is estimated to be, the sample size required

A) is smaller.

B) is larger.

C) is not affected.

D) can be smaller, larger or unaffected.

Answer: B

60) A confidence interval was used to estimate the proportion of statistics students who are females. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Based on the interval above, is the population proportion of females equal to 0.60?

A) No, and we are 90% sure of it.

B) No. The proportion is 54.17%.

C) Maybe. 0.60 is a believable value of the population proportion based on the information above.

D) Yes, and we are 90% sure of it.

Answer: C

61) The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, and she asked her assistant to use 25 days of data to construct the interval estimate, what confidence level can she attach to the interval estimate?

A) 99.7%

B) 99.0%

C) 98.0%

D) 95.4%

Answer: A

62) Which of the following would be an appropriate alternative hypothesis?

A) The mean of a population is equal to 55.

B) The mean of a sample is equal to 55.

C) The mean of a population is greater than 55.

D) The mean of a sample is greater than 55.

Answer: C

63) If a test of hypothesis has a Type I error probability (α) of 0.01, it means that

A) if the null hypothesis is true, you don’t reject it 1% of the time.

B) if the null hypothesis is true, you reject it 1% of the time.

C) if the null hypothesis is false, you don’t reject it 1% of the time.

D) if the null hypothesis is false, you reject it 1% of the time.

Answer: B

64) If a researcher does not reject a false null hypothesis, she has made a ________ error.

Answer: Type II

65) The symbol for the probability of committing a Type I error of a statistical test is

A) α.

B) 1 – α.

C) *β*.

D) 1 – *β*.

Answer: A

66) How many tissues should the Kimberly Clark Corporation package of Kleenex® contain? Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: = 52, S = 22. Suppose the alternative you wanted to test was *H*1: *μ* < 60. State the correct rejection region for α = 0.05.

A) Reject *H*0 if *t* > 1.6604.

B) Reject *H*0 if *t* < -1.6604.

C) Reject *H*0 if *t* > 1.9842 or *Z* < -1.9842.

D) Reject *H*0 if *t* < -1.9842.

Answer: B

67) If, as a result of a hypothesis test, you reject the null hypothesis when it is false, then you have committed

A) a Type II error.

B) a Type I error.

C) no error.

D) an acceptance error.

Answer: C

68) In testing for the differences between the means of two independent populations, you assume that the two populations each follow a ________ distribution.

Answer: normal

69) True or False: The sample size in each independent sample must be the same if we are to test for differences between the means of two independent populations.

Answer: FALSE

A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis. Assume that the two population variances are equal. A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes.

Gotham: G = 35 months, *S*G2 = 900 Metropolis: M = 50 months, *S*M2 = 1,050

70) Which of the following represents the relevant hypotheses tested by the real estate company?

A) *H*0: *μ*G – *μ*M ≥ 0 versus *H*1: *μ*G – *μ*M < 0

B) *H*0: *μ*G – *μ*M ≤ 0 versus *H*1: *μ*G – *μ*M > 0

C) *H*0: *μ*G – *μ*M = 0 versus *H*1: *μ*G – *μ*M ≠ 0

D) *H*0: G – M ≥ 0 versus *H*1: G – M < 0

Answer: A

71) What is the estimated standard error of the difference between the two sample means?

A) 4.06

B) 5.61

C) 8.01

D) 16.00

Answer: A

72) If we wish to determine whether there is evidence that the proportion of items of interest is the same in group 1 as in group 2, the appropriate test to use is

A) the *Z* test.

B) the *χ*2 test.

C) both A and B

D) neither A nor B.

Answer: C

73) True or False: A test for whether one proportion is higher than the other can be performed using the chi-square distribution.

Answer: FALSE

Four surgical procedures currently are used to install pacemakers. If the patient does not need to return for follow-up surgery, the operation is called a “clear” operation. A heart center wants to compare the proportion of clear operations for the four procedures, and collects the following numbers of patients from their own records:

They will use this information to test for a difference among the proportion of clear operations using a chi-square test with a level of significance of 0.05.

74) Referring to the table above, the test will involve ________ degrees of freedom.

Answer: 3

75) Referring to the table above, the overall or mean proportion of clear operations is ________.

Answer: 0.676

76) Referring to the table above, the expected cell frequency for the Procedure A/Clear cell is ________.

Answer: 25.69

77) Referring to Table the table above, the expected cell frequency for the Procedure D/Return cell is ________.

Answer: 5.83

78) Referring to the table above, the critical value of the test is ________.

Answer: 7.815

79) Referring to the table above, the value of the test statistic is ________.

Answer: 7.867

80) True or False: Referring to the table above, the null hypothesis will be rejected.

Answer: TRUE

81) True or False: Referring to the table above, the decision made suggests that the four procedures all have different proportions of clear operations.

Answer: FALSE

82) True or False: Referring to the table above, the decision made suggests that the four procedures do not all have the same proportion of clear operations.

Answer: TRUE

83) The *Y-intercept* (*b*0) represents the

A) estimated average *Y* when* X* = 0.

B) change in estimated average *Y* per unit change in *X*.

C) predicted value of *Y*.

D) variation around the sample regression line.

Answer: A

84) The strength of the linear relationship between two numerical variables may be measured by the

A) scatter plot.

B) coefficient of correlation.

C) slope.

D) *Y*-intercept.

Answer: B

85) True or False: When *r* = -1, it indicates a perfect relationship between *X* and *Y*.

Answer: TRUE

A computer software developer would like to use the number of downloads (in thousands) for the trial version of his new shareware to predict the amount of revenue (in thousands of dollars) he can make on the full version of the new shareware. Following is the output from a simple linear regression along with the residual plot and normal probability plot obtained from a data set of 30 different sharewares that he has developed: