Question

one cable company claims that it has excellent customer service. in fact, the company advertises that a technician will arrive within 45 minutes after a service call is placed. one frustrated customer believes this is not accurate, claiming that it takes over 45 minutes for the cable technician to arrive. the customer asks a simple random sample of 25 other cable customers how long it has taken for the cable technician to arrive when they have called for one. the sample mean for this group is 47.9 minutes with a standard deviation of 7.6 minutes. assume that the population distribution is approximately normal. test the customers claim at the 0.01 level of significance. step 3 of 3: draw a conclusion and interpret the decision. answer we reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance that it takes over 45 minutes for the cable technician to arrive. we reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance that it takes over 45 minutes for the cable technician to arrive. we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance that it takes over 45 minutes for the cable technician to arrive. we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance that it takes over 45 minutes for the cable technician to arrive.

Explanation:

Step1: Calculate the test - statistic

We use the one - sample t - test formula $t=\frac{\bar{x}-\mu}{s/\sqrt{n}}$, where $\bar{x} = 47.9$ (sample mean), $\mu = 45$ (hypothesized population mean), $s = 7.6$ (sample standard deviation), and $n = 25$ (sample size).
$t=\frac{47.9 - 45}{7.6/\sqrt{25}}=\frac{2.9}{7.6/5}=\frac{2.9}{1.52}\approx1.9079$

Step2: Determine the critical value

For a one - tailed test with $\alpha=0.01$ and degrees of freedom $df=n - 1=25 - 1 = 24$, the critical value from the t - distribution table is approximately $t_{0.01,24}= 2.492$.

Step3: Make a decision

Since the calculated t - value ($1.9079$) is less than the critical t - value ($2.492$), we fail to reject the null hypothesis.

Answer:

We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance that it takes over 45 minutes for the cable technician to arrive.