Is the national crime rate really going down? Some sociologists say yes! They sa

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. (Source: True Odds, by J. Walsh, Merritt Publishing.) Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 36 arrests last month, 19 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 in Rock Springs is different from 70%. STEP 1: What is the level of significance? STEP 2: State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? Ho: p = 0.7; H1: p > 0.7; right-tailed Ho: p = 0.7; H1: p 0.7; two-tailed Ho: p =0 .7; H1: p < 0.7; left-tailed Ho: p 0.7; H1: p = 0.7; two-tailed STEP 3: What sampling distribution will you use? Do you think the sample size is sufficiently large? The t distribution, since the sample size is large. The normal distribution, since sample size is large. STEP 4: What is the value of the sample test statistic? (Use 2 decimal places.) STEP 5: Find the P-value of the test statistic. (Use 4 decimal places.) STEP 6: Sketch the sampling distribution and show the area corresponding to the P-value. Maple Generated Plot Maple Generated Plot Maple Generated Plot Maple Generated Plot STEP 7: 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 ? At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. STEP 8: State your conclusion in the context of the application. Fail to reject the null hypothesis, there is insufficient evidence that the true proportion of arrests of males aged 14 to 34 in Rock Springs is different from 0.7. Fail to reject the null hypothesis, there is sufficient evidence that the true proportion of arrests of males aged 14 to 34 in Rock Springs is different from 0.7. Reject the null hypothesis, there is insufficient evidence that the true proportion of arrests of males aged 14 to 34 in Rock Springs is different from 0.7. Reject the null hypothesis, there is sufficient evidence that the true proportion of arrests of males aged 14 to 34 in Rock Springs is different from 0.7.

Explanation / Answer

STEP 1: What is the level of significance?

From the given,

alpha = 0.01 [ANSWER]

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STEP 2: State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

"test the claim that the population proportion of such arrests in Rock Springs is different from 70%" means we use a TWO TAILED TEST. Hence,

Ho: p = 0.7; H1: p 0.7; two-tailed [ANSWER]

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STEP 3: What sampling distribution will you use? Do you think the sample size is sufficiently large?

As n = 36 > 30, then

The normal distribution, since sample size is large. [ANSWER]

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STEP 4: What is the value of the sample test statistic? (Use 2 decimal places.)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   =   0.7
Ha:   p   =/=   0.7
As we see, the hypothesized po =   0.7      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.527777778      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.076376262      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -2.254918199   [ANSWER, TEST STATISTIC]

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5.

As this is a    2   tailed test, then, getting the p value,  
          
p =    0.024138464   [ANSWER]

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