Questions: A national online business magazine reports that the average cost of a speeding ticket in Miami, including court fees, is 220. A local police department claims that this amount has increased. To test their claim, they collect data from a simple random sample of 16 drivers who have been fined for speeding in the last year. Assuming that the distribution of speeding ticket costs is normally distributed and the population standard deviation is 17, is there sufficient evidence to support the police department's claim at the 0.05 level of significance? H0: μ=220 Ha: μ > 220

A national online business magazine reports that the average cost of a speeding ticket in Miami, including court fees, is 220. A local police department claims that this amount has increased. To test their claim, they collect data from a simple random sample of 16 drivers who have been fined for speeding in the last year. Assuming that the distribution of speeding ticket costs is normally distributed and the population standard deviation is 17, is there sufficient evidence to support the police department's claim at the 0.05 level of significance?

H0: μ=220
Ha: μ > 220

Transcript text: A national online business magazine reports that the average cost of a speeding ticket in Miami, including court fees, is $\$ 220$. A local police department claims that this amount has increased. To test their claim, they collect data from a simple random sample of 16 drivers who have been fined for speeding in the last year. Assuming that the distribution of speeding ticket costs is normally distributed and the population standard deviation is $\$ 17$, is there sufficient evidence to support the police department's claim at the 0.05 level of significance? \[ \begin{array}{l} H_{0}: \mu=220 \\ H_{a}: \mu > 220 \end{array} \]

Solution

Solution Steps

To test the claim, we need to perform a hypothesis test for the mean. The null hypothesis $ H_0 $ states that the average cost of a speeding ticket is $220, while the alternative hypothesis $ H_a $ suggests that the average cost is greater than $220. We will use a one-sample z-test since the population standard deviation is known.

Step 1: State the Hypotheses

The null hypothesis $ H_0 $ is that the average cost of a speeding ticket is $ \mu = 220 $. The alternative hypothesis $ H_a $ is that the average cost is greater than $ 220 $, i.e., $ \mu > 220 $.

Step 2: Calculate the Test Statistic

The sample mean is $ \bar{x} = 226.25 $, the population mean is $ \mu = 220 $, the population standard deviation is $ \sigma = 17 $, and the sample size is $ n = 16 $.

The z-score is calculated as: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} = \frac{226.25 - 220}{\frac{17}{\sqrt{16}}} = 1.4706 \]

Step 3: Determine the p-value

The p-value for a z-score of $ 1.4706 $ is $ 0.0707 $.

Step 4: Make a Decision

The significance level is $ \alpha = 0.05 $. Since the p-value $ 0.0707 $ is greater than $ \alpha $, we fail to reject the null hypothesis.