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 =	crimes per 1000 people
s =	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 60 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.

(d) Another neighborhood has a crime rate of 73 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.

(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

(f) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 60 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.

(g) Another neighborhood has a crime rate of 73 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.

(h) 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.

Yes. According to the central limit theorem, when n ? 30, the x distribution is approximately normal.Yes. According to the central limit theorem, when n ? 30, the x distribution is approximately normal.    No. According to the central limit theorem, when n ? 30, the x distribution is approximately normal.No. According to the central limit theorem, when n ? 30, the x distribution is approximately normal.

Answer 1

a) sample mean is

The sample standard deviation is

b) Uisng central limit theorem, we know that the distribution of sample mean is normal, with mean

The standard error of sample mean is

where n=46 is the sample size, and is the population standard deviation of crime rates in Denver neighborhoods.

But we do not know . We will estimate it using the sample standard deviation s. HEnce

The estimated standard error of sample mean is

The critical value for 80% confidence interval.

80% confidence interval indicates that the total area under both the tails is alpha=0.20. The area under the right tail is alpha/2= 0.20/2 = 0.10

the critcal value z which will give the right tail area of 0.10 is

this is same as

We can use the standard normal tables and get P(Z<1.28) = 0.5+0.3997 - 0.90

Hence the critical value is

80% confidence interval is

lower limit   =58.9	crimes per 1000 people
upper limit   =69.4	crimes per 1000 people

c) The crime rate in one of the neighborhoods is 60 per 1000

This value is within the 80% confidence interval of [58.9,69.4] that we have calculated. That means we can be 80% sure that the crime rate in this neighborhood is same as the average population crime rate. This means this rate is not below the average population crime rate and that fewer patrols could not safely be assigned to this neighborhood

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

d) Another neighborhood has a crime rate of 73 crimes per 1000 population. This value is higher than the upper limit of 80% confidence interval calculated in b.

That means we can be 80% sure that the crime rate in this neighborhood is not the same as the average population crime rate. This means that this crime rate seem to be higher than the population average and we would recommend assigning more patrols to this neighborhood.

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

e) 95% confidence interval indicates that the total area under 2 tails is alpha=0.05

The area under the right tail is alpha/2= 0.05/2 = 0.025

the critcal value z which will give the right tail area of 0.025 is

this is same as

We can use the standard normal tables and get P(Z<1.96) = 0.5+0.4750 - 0.975

Hence the critical value is

95% confidence interval is

the 95% confidence interval is

lower limit   =56.1	crimes per 1000 people
upper limit   =72.2	crimes per 1000 people

f) The crime rate in one of the neighborhoods is 60 per 1000

This value is within the 95% confidence interval of [56.1,72.2] that we have calculated. That means we can be 95% sure that the crime rate in this neighborhood is same as the average population crime rate. This means this rate is not below the average population crime rate and that fewer patrols could not safely be assigned to this neighborhood

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

g) Another neighborhood has a crime rate of 73 crimes per 1000 population. This value is higher than the upper limit of 95% confidence interval calculated in e.

That means we can be 95% sure that the crime rate in this neighborhood is not the same as the average population crime rate. This means that this crime rate seem to be higher than the population average and we would recommend assigning more patrols to this neighborhood.

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

Using central limit theorem we know that irrespective of the distribution of the population, the distribution of sample mean can be approximated by a normal distribution when the sample size n>30

Here we have a sample size n=46 which is greater than 30 and we are using the sample mean to draw conclusion about the population mean Hence we do not need to make any assumption regarding the distribution of population crime rates in Denver neighborhood.

ans: No. According to the central limit theorem, when n > 30, the distribution is approximately normal