The Department of Transportation would like to test the hypothesis that the average age of cars on the road is less than 12 years. A random sample of 45 cars had an average age of 10.6 years. It is believed that the population standard deviation for the age of cars is 4.1 years. The Department of Transportation would like to set α = 0.05. The p- value for this hypothesis test would be ______ .
A)0.1042
B)0.0110
C)0.0322
D)0.0788

Question

Quizplus · Accepted Answer

To find the p-value for this hypothesis test, we use the formula for the z-test since the population standard deviation is known. The formula for the z-score in this context is $z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean under the null hypothesis, $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values, we get $z = \frac{10.6 - 12}{4.1/\sqrt{45}}$. This calculation results in a z-score that corresponds to a p-value of 0.0110, indicating the probability of observing a sample mean of 10.6 years or less if the true average age of cars is 12 years. Given the significance level of $\alpha = 0.05$, a p-value of 0.0110 suggests that we reject the null hypothesis, supporting the claim that the average age of cars on the road is less than 12 years.

The Department of Transportation Would Like to Test the Hypothesis