# Can the process be “under control” in statistical terms but still fail to meet the needs of AnderSet’s customers? Explain, using a numerical example.

Question 1: (10 points)

Central Airlines would like to set up a control chart to monitor its on-time arrival performance. Each day over a 10-day period, Central Airlines chose 30 flights at random and tracked the number of late arrivals in each sample. The results are as follows:

Day Sample Size Number of late flights
1 30 2
2 30 3
3 30 4
4 30 0
5 30 1
6 30 6
7 30 4
8 30 2
9 30 3
10 30 5

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Set up a p chart to track the proportion of late arrivals.
Suppose the company takes two additional samples and finds that the number of late arrivals is 7 and 8 respectively. Has the process gone out of control?
Now suppose that the sample size is actually 50, not 30. Recalculate the control limits for the p chart. What happened? Explain
Question 2: (15 points)

AnderSet Laboratories produces rough lenses that will ultimately be ground into precision lenses for use in laboratory equipment. The company has developed the following thickness measures, based on 10 samples of five lenses that were taken when the process was under control:

Mean Minimum Maximum
3.9 3.617 3.989
4.206 3.971 4.302
4.214 4.062 4.4
3.89 3.749 3.937
4.036 3.501 4.084
4.134 3.543 4.584
3.037 2.935 3.929
5.082 3.797 5.695
3.404 2.837 4.255
5.246 5.106 6.382

Use these data to set up the appropriate control charts.
Can the process be “under control” in statistical terms but still fail to meet the needs of AnderSet’s customers? Explain, using a numerical example.
Suppose AnderSet Laboratories takes some additional samples of the same size, yielding the following results. Plot these samples on the control charts and circle any observations that appear to be out of control.
Mean Minimum Maximum
4.134 4.011 4.612
3.913 3.891 4.474
4.584 4.499 5.145
4.009 3.934 4.891
4.612 4.085 4.983
5.627 5.183 6.08

Question 3: (10 points)

VendAMovie rents newly-released movies through vending machines at gas stations. The company currently has a policy of having four DVDs of each movie they carry in stock. For a given title, on average three customers arrive per day to rent it and customers keep the DVD for a day. Assume that customer inter-arrival times and the amount of time they keep the DVD are exponentially distributed. Also, assume that due to lack of competitors at VendAMovie’s price point, customers are willing to wait for the DVD they would like to rent (you may assume that a customer who does not find the DVD in stock gets into a virtual queue by reserving it online).

On average, how long does a customer wait for a DVD?
VendAMovie estimates that the company incurs a loss of goodwill cost of \$1 per hour per waiting customer. If the hourly cost of keeping one DVD in the vending machine is \$1.3, should the company increase the number of DVDs to five?