Question

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 12.35%. Are older people more likely to be victimized

Answer 1

We conclude that older people are more likely to be victimized.

Explanation:

We are given that 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 12.35%

Let p = population proportion of people who are victimized.

So, Null Hypothesis,
`H_0` :
`p \leq` 10% {means that older people are less likely to be victimized or remains same}

Alternate Hypothesis,
`H_A` : p > 10% {means that older people are more likely to be victimized}

The test statistics that would be used here One-sample z-test for proportions;

T.S. =
`\frac{\hat p-p}{\sqrt{(p(1-p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of older people who are victimized = 12.35%

n = sample of older citizens = 527

So, the test statistics =
`\frac{0.1235-0.10}{\sqrt{(0.10(1-0.10))/(527) } }`

= 1.798

The value of z-test statistics is 1.798.

Since in the question, we are not given with the level of significance so we assume it to be 5%. Now at 5% level of significance, the z table gives a critical value of 1.645 for right-tailed test.

Since our test statistics is more than the critical value of z as 1.798 > 1.645, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that older people are more likely to be victimized.