Question

A cable company asserts that it provides outstanding customer service by ensuring a cable technician's arrival within 25 minutes after a service call is placed. A dissatisfied customer disputes this claim, arguing that it takes more than 25 minutes for the cable technician to arrive. To investigate, the customer conducts a simple random sample of 4 other cable customers and inquires about the time it took for the cable technician to arrive when they made a service call. The sample mean for this group is 29.6 minutes, with a standard deviation of 6.9 minutes. Assuming the population distribution is approximately normal, test the customer's claim at the 0.10 level of significance. State the null and alternative hypotheses for the test.

Answer 1

The customer tests the claim that the technician arrival time is within 25 minutes. The null hypothesis states the mean arrival time is 25 minutes or less, and the alternative hypothesis states it is greater. A one-sample t-test using the sample mean of 29.6 and standard deviation of 6.9 at 0.10 significance level will help validate the claim.

Step-by-step explanation:

The question involves conducting a hypothesis test to investigate a customer's claim regarding the time it takes for a cable technician to arrive after a service call. To do this, we state the null hypothesis (H0) and the alternative hypothesis (H1).

The null hypothesis is that the mean time for a technician to arrive is 25 minutes or less, while the alternative hypothesis is that the mean time is more than 25 minutes. Here are the hypotheses formulated mathematically:

H1: μ > 25

A sample size of 4 is very small, and this normally would be a limitation for many statistical tests. However, the assumption of an approximately normal population distribution of arrival times alleviates this concern somewhat. With a sample mean of 29.6 minutes and a standard deviation of 6.9 minutes, one would conduct a one-sample t-test because the population standard deviation is unknown and the sample size is less than 30.

Using the given sample statistics and level of significance (0.10), we can proceed to perform the t-test. If the t-statistic computed from this sample is greater than the critical t-value from the t-distribution with 3 degrees of freedom (n-1 = 4-1), we will reject the null hypothesis in favor of the alternative hypothesis. It's important to note that since the sample size is less than 30, the t-distribution is more appropriate than the z-distribution.