Refer to data in Exercise 7.3 on the labor time required to produce an order of automobile mufflers using a heavy stamping machine. The times (hours) for \(n=52\) orders of different parts has \(\bar{x}=1.865\) hours and \(s^{2}=1.5623\), so \(s=1.250\) hours.

(a) Conduct a test of hypotheses with the intent of showing that the mean labor time is more than 1.5 hours. Take \(\alpha=0.05\).

(b) Based on your conclusion in part (a), what error could you have made? Explain in the context of the problem.

Data From Exercise 7.3

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7.3 An industrial engineer collected data on the labor time required to produce an order of automobile mufflers using a heavy stamping machine. The data on times (hours) for n = 52 orders of different parts 2.15 2.27 0.99 0.63 2.45 1.30 2.63 2.20 0.99 1.00 1.05 3.44 0.49 0.93 2.52 1.05 1.39 1.22 3.17 0.85 1.18 2.27 1.52 0.48 1.33 4.20 1.37 2.70 0.63 1.13 3.81 0.20 1.08 2.92 2.87 2.62 1.03 2.76 0.97 0.78 4.68 5.20 1.90 0.55 1.00 2.95 0.45 0.70 2.43 3.65 4.55 0.33 = has 1.865 hours and s = 1.5623 so s = 1.250 hours. What can one assert with 95% confidence about the maximum error if x = 1.865 hours is used as a point estimate of the true population mean labor time required to run the heavy stamping machine?