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 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.

(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.

(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 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.

(g) 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.

(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 ) From excel we get sample mean = = 64.1 and sample standard deviation = s = 28.0

    Here we don't know popuation standard deviation , since we use t distribution to find confidence interval .

b ) Compute an 80% confidence interval for ?, the population mean crime rate for all Denver neighborhoods.

c = confidence level = 80%=0.80 and df = n - 1 = 46 - 1 = 45

So here critical value tc = 1.301 .

Confidence interval formula :

         Where ,

Lower limit =

Upper limit =

---------------------------------------------------------------------------

(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?

Confidence interval: (58.7 ,69.5 ) contains crime rate of 61 crimes per 1000 population.

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 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?

Confidence interval: (58.7 ,69.5 ) . crime rate of 75 crimes per 1000 population is above the upper limit .

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

------------------------------------------------------------------------

(e) Compute a 95% confidence interval for ?, the population mean crime rate for all Denver neighborhoods.

c = confidence level = 0.95 , df = 46 - 1 = 45 , so critical value =tc= 2.014

sample mean = = 64.1 and sample standard deviation = s = 28.0

Confidence interval :

      where ,

Lower limit :

Upper limit :

------------------------------------------------------------------------------------------

(f) 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?

crime rate of 61 crimes per 1000 population. is below the confidence lower limit .

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

--------------------------------------------------------------

g) 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?

crime rate of 75 crimes per 1000 population is above the upper limit of confidence interval .

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

-------------------------------------------------------------

Central Limit Theorem.:

The central limit theorem states that if you have a population with mean ? and standard deviation ? and take sufficiently large random samples from the population with replacement , then the distribution of the sample means will be approximately normally distributed.