An HR manager at a large agricultural equipment company is considering proposingputting the sales staff on 4 day - 10 hours/day work weeks, instead of their current 5day - 8 hours/day week. It is believed that this will decrease the amount of drivingthey must do during a typical work week. They put 16 of their sales people on the 4day week for one month, then on the 5 day week for one month, and have themrecord their driving distances. The data is recorded in the DrivingDistance dataset.Assume that all conditions for valid tests are met.
4-Day Week
2912
6112
6175
1102
3279
4997
1690
6606
1060
6392
3362
5555
1130
5771
8325
2376
5-Day Week
2798
7734
7505
844
4592
8117
1228
8728
1097
8099
3807
4128
989
6453
7012
3118
a. Test at the 5% level whether the 4 day work week shortens the driving distance.State the result of this test in terms of the data. Consider the confidence boundassociated with this test. Does it influence the decision to go to the 4-day week?Why or why not?
b. Compute the power of this test with a difference of 500. Does it influence thedecision?
c. Do a 2-sided test at the 5% level to test whether there is a statistically significantdifference in the driving distances. What is the probability of Type II error withthis test? Does this test influence the decision to go to the 4-day week? Whatfactors other than the data presented might the manager want to consider?
I know you use the Paired T test in minitab. Please explain each step.
To answer these questions using the paired t-test in Minitab, follow these steps:
Step 1: Input the data in Minitab.
- Open Minitab and navigate to "Stat" > "Basic Statistics" > "Paired t".
- Enter the data for the 4-day week in the first column and the data for the 5-day week in the second column.
- Alternatively, you can import the data from a file.
Step 2: Perform the paired t-test.
- In the Paired t-test dialog box, select the two columns containing the data.
- Click on "Options" to specify additional test settings, such as the confidence level and alternative hypothesis.
- Choose the desired confidence level (usually 95% or 0.05) and set the alternative hypothesis to "not equal".
- Click "OK" to perform the paired t-test.
Step 3: Interpret the results.
a. To test whether the 4-day work week shortens the driving distance at a 5% significance level:
- Look for the p-value in the output. If the p-value is less than the predetermined significance level (0.05), then there is a significant difference.
- In terms of the data given, if the p-value is less than 0.05, it means there is a statistically significant difference between the driving distances for the 4 and 5-day workweeks.
- The confidence bound associated with the test can be determined from the output. It represents the range within which we can be confident that the true mean difference in driving distances lies.
b. To compute the power of the test with a difference of 500:
- The power of a statistical test measures the probability of correctly rejecting a false null hypothesis.
- You can use an appropriate power analysis tool in Minitab to calculate the power of the test.
- Specify a difference of 500 as the effect size and use the sample sizes and standard deviation from the data to calculate the power.
- The power value obtained will indicate the likelihood of detecting a difference of 500 in driving distance between the two workweek schedules.
c. To perform a two-sided test to determine if there is a statistically significant difference in driving distances:
- Follow the same steps as in Step 2, but set the alternative hypothesis to "not equal".
- The p-value obtained from this test will indicate whether there is a significant difference in driving distances.
- The probability of Type II error (beta) is not directly provided by Minitab. However, it can be calculated as 1 - power. A higher Type II error probability means a greater likelihood of failing to reject a false null hypothesis.
Other factors to consider:
- Apart from the statistical results, the HR manager may also want to consider practical implications, such as employee satisfaction, work-life balance, and potential impact on productivity.
- Factors like potential cost savings, increased employee morale, and any logistical challenges associated with implementing the 4-day work week should also be taken into account before making a decision.