Question

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 $12, is there sufficient evidence to support the police departments claim at the 0.02 level of significance?
speeding ticket costs in miami
$225 $209 $234 $217 $226 $233 $240 $203
$233 $219 $220 $235 $229 $247 $213 $221
step 3 of 3: draw a conclusion and interpret the decision.
answer
we reject the null hypothesis and conclude that there is insufficient evidence at a 0.02 level of significance to support the claim that the average cost of a speeding ticket in miami has increased.
we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.02 level of significance to support the claim that the average cost of a speeding ticket in miami has increased.
we reject the null hypothesis and conclude that there is sufficient evidence at a 0.02 level of significance to support the claim that the average cost of a speeding ticket in miami has increased.
we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.02 level of significance to support the claim that the average cost of a speeding ticket in miami has increased.

Explanation:

Step1: State hypotheses

$H_0:\mu = 220$ (null hypothesis, where $\mu$ is the population - mean cost of a speeding ticket), $H_1:\mu>220$ (alternative hypothesis)

Step2: Calculate the test - statistic

First, calculate the sample mean $\bar{x}$. The sum of the data values: $225 + 209+234 + 217+226+233+240+203+233+219+220+235+229+247+213+221=3564$. The sample size $n = 16$, so $\bar{x}=\frac{3564}{16}=222.75$. The population standard deviation $\sigma = 12$. The test - statistic $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}=\frac{222.75 - 220}{\frac{12}{\sqrt{16}}}=\frac{2.75}{\frac{12}{4}}=\frac{2.75}{3}\approx0.92$.

Step3: Find the critical value

For a one - tailed test with $\alpha = 0.02$, the critical value $z_{\alpha}$ from the standard normal distribution table is approximately $2.05$.
Since the calculated $z = 0.92

Answer:

We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.02 level of significance to support the claim that the average cost of a speeding ticket in Miami has increased.