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, 25 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 =

State your conclusion in the context of the application.

-There is sufficient evidence at the 0.01 level to conclude that the true proportion of such arrests in the city differs from 70%.

-There is insufficient evidence at the 0.01 level to conclude that the true proportion of such arrests in the city differs from 70%.

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

-We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.

-We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.

-The conclusions obtained by using both methods are the same.

Answer 1

Solution :

Givan that ;

This is the two tailed test .

The null and alternative hypothesis is

H0 : p = 0.70

Ha : p 0.70

= x / n = 25 / 31 = 0.8065

P0 = 0.70

1 - P0 = 1 - 0.70 =0.30

= 0.01

Z = Z0.01 = + / - 2.58

The critical value = + / - 2.58

Test statistic = z

= - P0 / [P0 * (1 - P0 ) / n]

= 0.8065 - 0.70 / [0.70 ( 1 - 0.70 ) / 31 ]

= 1.293

The test statistic = 1.293

P-value = 0.1959

= 0.01

0.1959 > 0.01  

P-value <

Fail to reject the null hypothesis

Conclusion ; -    There is insufficient evidence at the 0.01 level to conclude that the true proportion of such arrests in the city differs from 70%.There is sufficient evidence to suggest that

Fail to reject p - value method