Question

The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.†

63.2	36.3	26.2	53.2	65.3	32.0	65.0
66.3	68.9	35.2	25.1	32.5	54.0	42.4
77.5	123.2	66.3	92.7	56.9	77.1	27.5
69.2	73.8	71.5	58.5	67.2	78.6	33.2
74.9	45.1	132.1	104.7	63.2	59.6	75.7
39.2	69.9	87.5	56.0	154.2	85.5	77.5
84.7	24.2	37.5	41.1

(1) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to one decimal place.)

mean:

sample standard deviation:

(2) Let us say the preceding data are representative of the population crime rates in Denver neighborhoods. Compute an 80% confidence interval for μ, the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.)

lower limit:

upper limit:
(3) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 56 crimes per 1000 population. Do you think that this rate is below the average population crime rate and that fewer patrols could safely be assigned to this neighborhood? Use the confidence interval to justify your answer.

a) Yes. The confidence interval indicates that this crime rate is below the average population crime rate.

b) Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

c) No. The confidence interval indicates that this crime rate is below the average population crime rate.

d) No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(4) Another neighborhood has a crime rate of 77 crimes per 1000 population. Does this crime rate seem to be higher than the population average? Would you recommend assigning more patrols to this neighborhood? Use the confidence interval to justify your answer.

a) Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

b) Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.    

c) No. The confidence interval indicates that this crime rate is higher than the average population crime rate.

d) No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(5) Compute a 95% confidence interval for μ, the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.)

lower limit:

upper limit:

(6) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 56 crimes per 1000 population. Do you think that this rate is below the average population crime rate and that fewer patrols could safely be assigned to this neighborhood? Use the confidence interval to justify your answer.

a) Y

es. The confidence interval indicates that this crime rate is below the average population crime rate.

b) Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.    

c) No. The confidence interval indicates that this crime rate is below the average population crime rate.

d) No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(7) Another neighborhood has a crime rate of 77 crimes per 1000 population. Does this crime rate seem to be higher than the population average? Would you recommend assigning more patrols to this neighborhood? Use the confidence interval to justify your answer.

a) Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

b) Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.    

c) No. The confidence interval indicates that this crime rate is higher than the average population crime rate

d) No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(8) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

a) Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

b) Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.    

c)No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

d) No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

Answer 1

(1)

sample mean x = 64.2

sample standard deviation s = 27.9

(2)

Degree of freedom = n-1 = 46-1 = 45

Critical value of t at 80% confidence interval and df = 45 is 1.30

Standard error of mean = s / = 27.9 / = 4.11

lower limit: 64.2 - 1.3 * 4.11 = 58.9

upper limit: 64.2 + 1.3 * 4.11 = 69.5

(3)

Since the lower limit of the confidence interval is greater than 56,

a) Yes. The confidence interval indicates that this crime rate is below the average population crime rate.

(4)

Since the upper limit of the confidence interval is less than 77,

b) Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.    

(5)

Critical value of t at 95% confidence interval and df = 45 is 2.01

lower limit: 64.2 - 2.01 * 4.11 = 55.9

upper limit: 64.2 + 2.01 * 4.11 = 72.4

(6)

Since the 95% confidence interval contains the value 56,

d) No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(7)

Since the upper limit of the 95% confidence interval is less than 77,

b) Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.    

(8)

a) Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.