Solved – Problem understanding what type of test to use and how to proceed with the questions provided

hypothesis testingself-study

I have a case study where a company made new golf balls with stronger coatings than current golf balls. One of the technicians is concerned that the new coating may decrease the driving distance. An experiment was conducted in which 40 types of each ball was hit by a machine to determine mean driving distance of the two balls. The distances for each ball were given in a spreadsheet. Here are the samples: 

Current: 264.00 261.00 267.00 272.00 258.00 283.00 258.00 266.00 259.00 270.00 263.00 
         264.00 284.00 263.00 260.00 283.00 255.00 272.00 266.00 268.00 270.00 287.00 
         289.00 280.00 272.00 275.00 265.00 260.00 278.00 275.00 281.00 274.00 273.00 
         263.00 275.00 267.00 279.00 274.00 276.00 262.00
New;     277.00 269.00 263.00 266.00 262.00 251.00 262.00 289.00 286.00 264.00 274.00 
         266.00 262.00 271.00 260.00 281.00 250.00 263.00 278.00 264.00 272.00 259.00 
         264.00 280.00 274.00 281.00 276.00 269.00 268.00 262.00 283.00 250.00 253.00 
         260.00 270.00 263.00 261.00 255.00 263.00 279.00

These are the questions:

  1. Formulate and present the rationale for a hypothesis test that Par could use to compare the driving distances of the current and new golf balls
  2. The CEO tells you (in contrast to what Bill said) that he would like to sell the new ball unless there is overwhelming evidence that the cut‐resistant ball is slower than the old ball so you should frame your hypothesis accordingly. Hint: you are more willing to make a Type II error. Analyze the data to provide the hypothesis testing conclusion and statements (at .05). Also include narrative versions of what ever of the above forms you use such as: Ho: The mean is … What is the p‐value for your test? What is your recommendation for Par, Inc.?
  3. Provide the following descriptive summaries of the data for each model: ONLY the means and variances and the z values.
  4. What is the 95% confidence interval for the difference between the means of the two populations?
  5. Do you see a need for larger sample sizes and more testing with the golf balls? Discuss.

I've looked for help elsewhere and everyone seems to use the t-test to conduct the analysis. Shouldn't the z-test be used since the sample sizes are both over 30? I would also like some direction in understanding how to answer the other questions but my main question is the most important.

Best Answer

If we can assume the driving distances are normally distributed with possibly different means but a common variance then the t test is the exact best test. A sample size of 30 just means that the t distribution is getting close to the normal so the z test approximates the t test. But that does not mean it should be preferred. Sometimes it is used on the grounds that it is simpler. If the two variances differ you have the famous Behrens-Fisher problem and the normalized distribution of the mean difference is neither normal nor t. However there is an approximate t distribution with non-integer degrees of freedom that approximates the null distribution of the test statistic. If the two population distributions are very non-normal and the sample size is not large a test of the null hypothesis that the two distributions are the same versus that they come from different distributions with different medians then a nonparametric test is appropriate. The Wilcoxon rank sum test would be preferred.

If the data are paired there is a paired t test in the first situation and a signed rank test in the last case.

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