Question

Suppose employees at a company are unionized. In the next round of negotiations regarding the collective agreement, the union wants to increase the expected amount of time to complete a particular task to 2 minutes. The union claims that the time to complete this task follows a Normal distribution with a mean of 1 minute and a standard deviation of 30 seconds, so a limit of 2 minutes would cover roughly 95% of task times, which the union feels is fair.

The union and the company agree to allow an independent assessor to time how long it takes to complete the task. The assessor watches the task being completed 10 times, and obtains the follow data set.

Index |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |

Time (minutes) |
1.2 |
1.5 |
1.75 |
2.0 |
2.5 |
2.75 |
3 |
1.5 |
0.5 |
1 |

(Recall an index is just a way to label which time is which, so
for example, on the 5^{th} time they observed the task, it
took 2.5 minutes to complete).

a) [6 marks] What is the probability that the assessor observes a sample mean task time as high as the one they see, or even higher, if the population mean time to complete the task really is 1 minute, as the union claims? Hint: you’ll first have to calculate the sample mean. Make sure to show all of your work in this question, just like in the last question.

c) [2 marks] Based on your result from part b), do you believe the union’s claim that the population mean time to complete the task is 1 minute? Why or why not?

Answer #1

The sum of given observations is 17.7 minutes. Hence, the observed mean is 1.77 minutes against the claim of 1 minute and a standard deviation of 0.5 minute.

The probability to obsrve this mean task time is obtained by first finding its Z value, and then using Z table to observe a mean value greater than this.

This is the observed statistic based on one-tail hypothesis test that the mean task time is 1 minute about 95% of the times. So against the critical value of 0.05, the p-value is 0.0618. Since observed statistic is more than the critical value, hence we fail to reject the null hypothesis and accept that the mean task time is 1 minute.

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