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 34 arrests last month, 21 were of males aged 15 to 34 years. Use a 10% 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? Incorrect with credit: Your answer is incorrect, but you received credit for a previous answer. State the null and alternate hypotheses. H0: p < 0 .7; H1: p = 0.7 H0: p ≠ 0.7; H1: p = 0.7 H0: p = 0.7; H1: p > 0.7 H0: p = 0 .7; H1: p < 0.7 H0: p = 0.7; H1: p ≠ 0.7

(b) What sampling distribution will you use? 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. The Student's t, since np < 5 and nq < 5. Incorrect with credit: Your answer is incorrect, but you received credit for a previous answer. 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.)

(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 = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.10 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 0.10 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 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.

Answer 1

(a)

The level of significance = = 0.10

Correct option:

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

(b)

Correct option:

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

n = 34

= 21/34 = 0.6176

So,

= 1 - 0.6176 = 0.3824

So,

Test statistic is given by:

So,

Sample Test Statistic is:

- 0.99

(c)

Table of Area Under Standard Normal Curve gives area = 0.3389

So,

P - value = (0.5 - 0.3389) X 2

          = 0.3222

So,

P - value for the test statistic is 0.3222

(d)

Correct option:

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e)

Correct option:

There is insufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.