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 (a) 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.) x = 64.16 Correct: Your answer is correct. crimes per 1000 people s = 27.86 Correct: Your answer is correct. crimes per 1000 people (b) Let us say the preceding data are representative of the population crime rates in Denver neighborhoods. Compute an 80% confidence interval for ?, the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.) lower limit crimes per 1000 people upper limit crimes per 1000 people (c) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 61 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. Yes. The confidence interval indicates that this crime rate is below the average population crime rate. Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate. No. The confidence interval indicates that this crime rate is below the average population crime rate. No. The confidence interval indicates that this crime rate does not differ from the average population crime rate. Correct: Your answer is correct. (d) Another neighborhood has a crime rate of 75 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. Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate. Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate. No. The confidence interval indicates that this crime rate is higher than the average population crime rate. No. The confidence interval indicates that this crime rate does not differ from the average population crime rate. Correct: Your answer is correct. (e) Compute a 95% confidence interval for ?, the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.) lower limit crimes per 1000 people upper limit crimes per 1000 people

Answer 1

Solution:

a. Sample mean, x-bar = 64.16

Using excel, =AVERAGE function we get

x-bar = 64.16

Sample standard deviation, s = 27.86

Using excel, =stdev.s function we get

s = 27.86

b. First, we compute the t-score at a= 1 - 0.80 = 0.20 and degrees of freedom, df = n - 1 = 46 - 1 = 45

t (a/2, df) = t (0.10, 45) = 1.301

80% confidence interval is given by:-

64.16 1.301*27.86/46

64.16 1.301*4.108

64.16 5.34

58.8, 69.5

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

Since 61 crimes lies within the confidence interval from 58.8 to 69.5, the crime rate does not differ from the average population crime rate.

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

Since 75 crimes lies outside the confidence interval from 58.8 to 69.5, the crime rate is higher than the average population rate.

e. 95% confidence interval is given by:-

64.16 t (0.025, 45)*27.86/46

64.16 2.014*27.86/46

64.16 2.014*4.108

64.16 8.27

55.9, 72.4