A pipes manufacturer makes pipes with a length that is supposed to be 10 inches. A quality control technician sampled 18 pipes and found that the sample mean length was 10.07 inches and the sample standard deviation was 0.27 inches. The technician claims that the mean pipe length is not 10 inches. What type of hypothesis test should be performed? Select What is the test statistic? Ex: 0.123 What is the number of degrees of freedom? Ex: 20 Does sufficient evidence exist at the $\alpha=0.01$ significance level to support the technician's claim?

Step 1 ：The technician is making a claim about the population mean, so we should perform a t-test for a single population mean. The null hypothesis is that the population mean is 10 inches, and the alternative hypothesis is that the population mean is not 10 inches. This is a two-tailed test because the technician is not specifying whether the mean is greater than or less than 10 inches.

Step 2 ：The test statistic can be calculated using the formula for the t-statistic: \(t = \frac{\bar{x} - \mu}{s/\sqrt{n}}\) where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean under the null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 3 ：Given that the sample size \(n = 18\), the sample mean \(\bar{x} = 10.07\), the population mean under the null hypothesis \(\mu = 10\), and the sample standard deviation \(s = 0.27\), we can substitute these values into the formula to get the test statistic \(t = 1.099943881845745\).

Step 4 ：The degrees of freedom for a t-test is \(n - 1\), so in this case, the degrees of freedom is \(df = 17\).

Step 5 ：To determine whether there is sufficient evidence to support the technician's claim, we will compare the p-value of the test statistic to the significance level. If the p-value is less than the significance level, we reject the null hypothesis and conclude that there is sufficient evidence to support the technician's claim. The p-value in this case is \(p = 0.2866892479295742\), and the significance level is \(\alpha = 0.01\).

Step 6 ：Since the p-value is greater than the significance level, we fail to reject the null hypothesis. Therefore, there is not sufficient evidence at the \(\alpha=0.01\) significance level to support the technician's claim that the mean pipe length is not 10 inches.

Step 7 ：\(\boxed{\text{Final Answer: The type of hypothesis test that should be performed is a t-test for a single population mean. The test statistic is approximately 1.10. The number of degrees of freedom is 17. There is not sufficient evidence at the }\alpha=0.01\text{ significance level to support the technician's claim. Therefore, we fail to reject the null hypothesis that the mean pipe length is 10 inches. The p-value of approximately 0.29 is greater than the significance level of 0.01, indicating that the observed data is not significantly different from what would be expected under the null hypothesis.}}\)