Question

You are on the police force in a small town. During an election year, a candidate for mayor claims that fewer police are needed because the average police officer makes only four arrests per year. You think the population mean is much higher than that, so you conduct a small research project. You ask 12 other officers how many arrests they made in the past year. The average for this sample of 12 is 6.3, with a standard deviation of 1.5. With your sample evidence, test the null hypothesis that the population mean is four arrests against the alternative that it is greater than four. Set your alpha level at .01.

Answer 1

Let be the true average number of arrests per year made by an officer of the town. We want to test the null hypothesis that the population mean is four arrests against the alternative that it is greater than four

The hypotheses are

The significance level to test the hypothesis is

We know the following

n=12 is the sample size

is the sample average number of arrests

is the sample standard deviation of the number of arrests

We do not know the population standard deviation of the number of arrests. We will use the sample to estimate the population SD

The estimated population standard deviation of number of arrests in the town is

The estimated standard error of mean is

The hypothesized value of mean number of arrests is (from the null hypothesis)

The sample size is less than 30 and we do not know the population standard deviation. Assuming that the number of arrests is normally distributed, using the central limit theorem, we say that the sampling distribution of mean is t distribution.

Hence we use a t test.

the test statistic is

This is a right tail (one tail) test (The alternative hypothesis is ">")

The right tail critical value of t is obtained using

The degrees of freedom for t are n-1=12-1=11

Using the t tables for df=11 and area under the right tail=0.01 (or the combined area under 2 tails =0.02), we get

The critical value is 2.718

We will reject the null hypothesis, if the test statistic is greater than the critical value.

Here, the test statistic is 5.312 and it is greater than the critical value 2.718. Hence we reject the null hypothesis.

We conclude that there is sufficient evidence to support the claim that the population mean is greater than four arrests.