Question

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years.† Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 31 arrests last month, 24 were of males aged 15 to 34 years. Use a 1% level of significance to test the claim that the population proportion of such arrests is the city different from 70%. Solve the problem using both the traditional method and the P-value method. Since the sampling distribution of p̂ is the normal distribution, you can use critical values from the standard normal distribution as shown in the table of critical values of the z distribution. (Round the test statistic and the critical value to two decimal places. Round the P-value to four decimal places.)

test statistic	=
critical value	= ±
P-value	=

Answer 1

Solution:

Here, we have to use z test for population proportion.

H₀: p = 0.7 versus H_a: p ≠ 0.7

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 24

n = sample size = 31

p̂ = x/n = 24/31 = 0.774193548

p = 0.7

q = 1 - p = 0.3

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.774193548 – 0.7)/sqrt(0.7*0.3/31)

Z = 0.9014

Test statistic = 0.90

Critical value = - 2.58 and 2.58

P-value = 0.3674

(by using z-table)

Test statistic is lies between critical values.

P-value > α = 0.01

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that the population proportion of such arrests is the city different from 70%.