Question

A state Department of Transportation claims that the mean wait time for various services at its different locations is at most 6 minutes. A random sample of 34 services at different locations has a mean wait time of 10.3 minutes and a standard deviation of 8.0 minutes. Is there enough evidence to reject the claim at alpha equals 0.01? Find the critical value. Round your answer to the thousandths place.

Answer 1

To determine whether there is enough evidence to reject the claim of the state Department of Transportation, a hypothesis test needs to be performed. The null hypothesis (H0) is that the mean wait time is 6 minutes or less, while the alternative hypothesis (H1) is that the mean wait time is greater than 6 minutes. Using a one-sample t-test, the test statistic is calculated and compared to the critical value obtained from the t-distribution table or calculator. In this case, the test statistic is greater than the critical value, indicating that there is enough evidence to reject the null hypothesis.

Step-by-step explanation:

To determine whether there is enough evidence to reject the claim of the state Department of Transportation, we need to perform a hypothesis test.

Hypotheses:

Let:

μ be the mean wait time for various services at different locations.

The null hypothesis (H0) is that the mean wait time (μ) is 6 minutes or less:

H0: μ ≤ 6
The alternative hypothesis (H1) is that the mean wait time (μ) is greater than 6 minutes:

H1: μ > 6

Test statistic:

We will use a one-sample t-test since the population standard deviation is unknown.

The test statistic (t) is calculated using the formula:

t = (Xbar - μ)/(s/√(n))

Where:

Xbar is the sample mean - 10.3 minutes,

μ is the claimed mean - 6 minutes,

s is the sample standard deviation - 8.0 minutes, and

n is the sample size - 34.

Critical value:

To find the critical value, we need to determine the rejection region using the significance level (alpha) of 0.01 and the degrees of freedom (df) which is n - 1 = 33.

From the t-distribution table or calculator, the critical value for a one-tailed test at alpha = 0.01 and df = 33 is approximately 2.449.

Conclusion:

The test statistic (t) = (10.3 - 6)/(8.0/√(34)) = 3.720.

Since the test statistic (t) = 3.720 > 2.449 (the critical value), we have enough evidence to reject the null hypothesis.

Therefore, we can conclude that there is sufficient evidence to reject the claim of the state Department of Transportation at alpha = 0.01.