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 39 arrests last month, 29 were of males aged 15 to 34 years. Use a 5% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀: p = 0.7; H₁: p ≠ 0.7H₀: p < 0 .7; H₁: p = 0.7    H₀: p = 0.7; H₁: p > 0.7H₀: p = 0 .7; H₁: p < 0.7H₀: p ≠ 0.7; H₁: p = 0.7

(b) What sampling distribution will you use?

The Student's t, since np > 5 and nq > 5.The standard normal, since np > 5 and nq > 5.    The Student's t, since np < 5 and nq < 5.The standard normal, since np < 5 and nq < 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.There is insufficient evidence at the 0.05 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.    

Answer 1

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 39 arrests last month, 29 were of males aged 15 to 34 years. Use a 5% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

(a) What is the level of significance?

Answer: The level of significance is 5%.

State the null and alternate hypotheses.

Answer: H₀: p = 0.7; H₁: p ≠ 0.7

Reason: We want to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.So the null hypothesis is that proportion is 0.7 while alternate hypothesis is that it is different than 0.7.

(b) What sampling distribution will you use?

Answer: The standard normal, since np > 5 and nq > 5.

Reason:

Sample size = n = 39 , p = 0.7 and q = 1 - 0.7 = 0.3.

np = 39*0.7 = 27.3 > 5

nq = 39*0.3 = 11.7 > 5

What is the value of the sample test statistic? (Round your answer to two decimal places.)

Answer: sample statistic Z = 0.5942

Calculations:

The test statistic, z is ggive by following formula:

where, p₀ = 0.7,

p = 29/39 = 0.7436

n=39

z = 0.5942

(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)

Answer: P-value = 0.55

Calculations:

P-value = P( Z < - 0.5942) + P(Z > 0.5942)

= 2 * P(Z<-0.5942)

= 2* 0.2762

=0.5524

The sampling distribution of Z is shown in the figure below with the area corresponding to P-value, which is in the shaded region

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

Answer: At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

Reason: As, p-value > α = 0.05, the null hypothesis cannot be rejected. Therefore, the data is not statitically significant.

(e) Interpret your conclusion in the context of the application.

Answer: There is insufficient evidence at the 0.05 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.

Reason: As the null hypothesis is not rejected, we conclude that there is not significant evidence to suggest the alternate hypothesis is true.