   Chapter 8, Problem 8.11P

Chapter
Section
Textbook Problem

C J About half of the police officers in Shinbone, Kansas, have completed a special course in investigative procedures. Has the course increased their efficiency in clearing crimes by arrest? The proportions of cases cleared by arrest for samples of trained and untrained officers are reported here. Sample 1 (Trained) Sample 2 (Untrained) P s 1 = 0.47 P s 2 = 0.43 N 1 = 157 N 2 = 113

To determine

To find:

If the course has increased the efficiency of police officers in clearing crimes by arrest.

Explanation

Given:

The sample statistics is given in the table below,

 Sample 1(Trained) Sample 2(Untrained) Ps1=0.47 Ps2=0.43 N1=157 N2=113

Approach:

The five step model for hypothesis testing is,

Step 1. Making assumptions and meeting test requirements.

Step 2. Stating the null hypothesis.

Step 3. Selecting the sampling distribution and establishing the critical region.

Step 4. Computing test statistics.

Step 5. Making a decision and interpreting the results of the test.

Formula used:

The formula to calculate the sampling distribution of the differences in sample proportions of large samples is given by,

Z(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Where, Ps1 and Ps2 is the proportion of first and second sample respectively,

Pu1 and Pu2 is the proportion of first and second population respectively,

σpp is the standard deviation and the formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

Where, N1 and N2 is the number of first and second population respectively.

And Ps1 is the population proportion and the formula to Ps1 is given by,

Pu=N1Ps1+N2Ps2N1+N2

Calculation:

As the significant difference in the sample proportions is to be determined and a direction is predicted, a one tailed test is applied.

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Independent random samples.

Level of measurement is nominal.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

The statement of the null hypothesis is that there is no significant difference in the samples of the population. Thus, the null and the alternative hypotheses are,

H0:Pu1=Pu2

H1:Pu1>Pu2

Step 3. Selecting the sampling distribution and establishing the critical region.

Since, the sample size is large, Z distribution can be used.

Thus, the sampling distribution is Z distribution.

The level of significance is,

α=0.05

Area of critical region is,

Z(critical)=±1.65

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate Pu is given by,

Pu=N1Ps1+N2Ps2N1+N2

Substitute 0.47 for Ps1, 0.43 for Ps2, 157 for N1, and 113 for N2 in the above mentioned formula,

Pu=157×0

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