   Chapter 7, Problem 7.11P

Chapter
Section
Textbook Problem

S O C / C J A survey shows that 10% of the population is victimized by property crime each year. A random sample of 527 older citizens (65 years or more of age) shows a victimization rate of l4%. Are older people more likely to be victimized? Conduct both one- and two-tailed tests of significance.

To determine

To find:

The significant difference between the given statements.

Explanation

Given:

The given information is,

A survey shows that 10% of the population is victimized by property crime each year. A random sample of 527 older citizens (65 years or more of age) shows a victimization rate of l4%.

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:

For large samples with single sample proportions, the Z value is given by,

Z(obtained)=PsPuPu(1Pu)/N

Where, Ps is the sample proportion,

Pu is the population proportion and,

N is the sample size.

Calculation:

From the given information, the sample size is 527, sample proportion is 14% and the population proportion is 10%.

Follow the steps for one-tailed hypothesis testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Random sampling.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

The statement of the null hypothesis is that older citizens are not likely to be victimized more than the population average and the alternative hypothesis is that older citizens are more likely to be victimized than the population average.

Thus, the null and the alternative hypotheses are,

H0:Pu=0.10

H1:Pu>0.10

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

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

The information given is in the form of proportions, the test for proportion can be used.

Thus, the sampling distribution is Z distribution.

The level of significance is α=0.05.

And area of critical region is Z(critical)=1.645.

Step 4. Computing test statistics.

For large samples with single sample proportions, the Z value is given by,

Z(obtained)=PsPuPu(1Pu)/N ……(1)

For sample proprtion Ps=0.14, population proportion Pu=0.10 and sample size N=527, the Z value is given by,

Substitute 0.14 for Ps, 0.10 for Pu and 527 for N in equation(1).

Z(obtained)=0.140.100

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