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 yearst. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 37 arrests last month, 25 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? 0.05 State the null and alternate hypotheses. 0 Ho: p = 0.7; F:P = 0.3 • Hs: P = 0.7, H.: p = 0.7 Ho: p=0.7; H4: P<0.7 0 Fo: P = 0.7; H: p = 0.3 - Ho: p < 0.3; H:P = 0.7 (b) What sampling distribution will you use? The Student's t, since np < 5 and ng < 5. The standard normal, since np > 5 and ng > 5. The standard normal, since np < 5 and ng < 5. The Student's t, since np > 5 and ng > 5. What is the value of the test statistic? (Round your answer to two decimal places.) -0.32 (c) Find the p-value of the test statistic. (Round your answer to four decimal places.) 0.7490 Sketch the sampling distribution and show the area corresponding to the p-value,

Answer 1

The level of significance is $\alpha =0.05$ .

We have to test

$H_0:p=0.7\\ H_1:p\neq 0.7$

b) We use Standard normal distribution since

The estimated proportion is $\widehat{p}=25/37=0.6757$

The test statistic is

$z=\frac{\widehat{p}-p_0}{\sqrt{\frac{\widehat{p}\left ( 1-\widehat{p}\right )}{n}}}\\ z=\frac{0.6757-0.7}{\sqrt{0.6757 \left ( 1-0.6757\right )/37}}\\ {\color{Blue} z=-0.32}$

c)The P-value of the test is

$\textup{p-value}=2\Phi (-0.32)\\ {\color{Blue} \textup{p-value}=0.7490}$

The third graph is the correct graph.

d)Since the P-value is $\textup{P-value}=0.749>\alpha =0.05$ , we accept the null hypothesis. Strong evidence to support Null hypothesis.

The conclusion is : " at a 5% level of significance we fail to reject the null hypothesis and conclude that the data are not statistically significant"

The last choice is correct.