Question

A simple random sample of 40 recorded speeds (in mi/h) is obtained from cars traveling on a specific highway. The sample has a mean of 68.4 mi/h and a standard deviation of 5.7 mi/h. Use a 0.05 significance level to test the claim that the mean speed of all cars is greater than the posted speed limit of 65 mi/h. State the initial and final conclusion.

A. Reject the null hypothesis; there is not sufficient evidence to warrant a rejection of the claim that the mean speed is greater than 65 mi/h.
B. Fail to reject the null hypothesis; there is sufficient evidence to warrant a rejection of the claim that the mean speed is less than 65 mi/h.
C. Fail to reject the null hypothesis; there is not sufficient evidence to warrant a rejection of the claim that the mean speed is less than 65 mi/h.
D. Reject the null hypothesis; there is sufficient evidence to support the claim that the mean speed is greater than 65 mi/h.

Answer 1

To test the claim that the mean speed of all cars is greater than the posted speed limit of 65 mi/h, we perform a one-sample t-test. The test statistic is calculated using the sample mean, hypothesized mean, sample standard deviation, and sample size, and compared to the critical value. We reject the null hypothesis if the test statistic is greater than the critical value.

Step-by-step explanation:

To test the claim that the mean speed of all cars is greater than the posted speed limit of 65 mi/h, we will perform a one-sample t-test. The null hypothesis (H0) is that the mean speed is equal to 65 mi/h, and the alternative hypothesis (Ha) is that the mean speed is greater than 65 mi/h.

Since the sample size is large (40) and the population standard deviation is unknown, we will use the t-distribution. The test statistic for a one-sample t-test is calculated as:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))

Using the given information, the test statistic is:

t = (68.4 - 65) / (5.7 / sqrt(40)) = 2.597

Next, we need to find the critical value at the 0.05 significance level using the t-distribution with 39 degrees of freedom (40 - 1). The critical value for a one-tailed test is approximately 1.684. Since our test statistic (2.597) is greater than the critical value, we reject the null hypothesis.

The final conclusion is: Reject the null hypothesis; there is sufficient evidence to support the claim that the mean speed is greater than 65 mi/h.