What hypothesis testing procedure should you use to answer these questions? Assume all samples are simple random samples, and assume α = 0.05 if it isn’t specified.
A company wishes to purchase a new machine to produce ball bearings since the new machine is faster, and so will produce more ball bearings than the old machine. They decide to purchase the new machine only if the mean strength of the bearings is not less than the old machine. The old machine produces bearings that can withstand 400 lbs per square inch. The company is allowed to test the new machine for a day, so they produce 100 ball bearings under the usual operating conditions on the new machine. They find that the average strength to be approximately 398.5 lbs per square inch and the SD to be about 5 lbs per square inch. The sample distribution does not look normal. Should the company purchase the new machine, at α = 0.05?
In this scenario, the company wants to compare the mean strength of the ball bearings produced by the new machine (sample) to the mean strength of the ball bearings produced by the old machine (population parameter).
Since the sample distribution does not look normal, we can use a non-parametric hypothesis testing procedure called the Mann-Whitney U test to compare the two groups. The Mann-Whitney U test is appropriate for comparing two independent groups when the data do not meet the assumptions of a parametric test.
The null hypothesis for this test would be: "The mean strength of the ball bearings produced by the new machine is less than or equal to the mean strength of the ball bearings produced by the old machine."
The alternative hypothesis would be: "The mean strength of the ball bearings produced by the new machine is greater than the mean strength of the ball bearings produced by the old machine."
To perform the Mann-Whitney U test, you would calculate the U statistic and compare it to critical values from the Mann-Whitney U distribution. If the calculated U falls in the rejection region, you would reject the null hypothesis and conclude that the mean strength of the ball bearings produced by the new machine is significantly greater than the mean strength of the ball bearings produced by the old machine.
Note: It is important to double-check the assumption about the sample distribution not being normal. If the data is approximately normal or can be transformed to be approximately normal, you could consider using a t-test instead.