Question

Progressive is investigating its claim processing time. Table below shows the samples that have been taken from 3 branches for 15 random days. Examine the process output by calculating the control limits and developing X-bar and R control charts using the data given. Does the process look to be in statistical control?

Day	Branch 1	Branch 2	Branch 3
1	14	16	27
2	30	13	23
3	18	25	14
4	13	21	19
5	14	10	12
6	18	22	21
7	9	11	18
8	17	12	19
9	19	17	21
10	10	8	13
11	12	19	20
12	16	10	12
13	13	15	29
14	15	8	14
15	21	19	18

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Answer 1

Given data is as follows:

The sample mean and range is calculated as follows:
The Sample mean for set 1 = Sum of all the samples for set 1/Number of samples in set 1 =57/3 = 19
The Range for day 1 = Maximum value of all the samples in set 1 - Minimum value of all samples in set 1 = 27 -14 = 13

The details for all the sets is as follows:

Calculations for UCL, LCL and X bar chart are as follows:
Suppose we have k subgroups, each of size n. Let xij represent the measurement in the jth sample of the ith subgroup.

The ith subgroup mean is calculated using ;

and was calculated previously.

The grand mean (x-bar-bar) is calculated as follows:

= 18.133

Also, the following is another formula for the calculation of grand mean (x-bar-bar):

The range of the above data is calculated using the formula below:

= 9.733

Also for the sample size of n = 3, we have the following values of A2, D2, D3, and D4 is as follows from the table of control chart constants.

The following are the calculations for UCL, LCL and centreline for X-bar chart
UCL (X-bar) = X-bar-bar + (A2 x R-bar) = 18.1333 + 1.0232 * 9.7333 = 28.0925
LCL (X-bar) = X-bar-bar - (A2 x R-bar) = 18.1333 - 1.0232 * 9.7333 = 8.1742
Centreline of X-bar chart is X-bar-bar (grand mean) = 18.1333
The following are the calculations for UCL, LCL and centreline for R chart
UCL (R) = R-bar * D4 = 9.7333 * 2.574 = 25.0536
LCL (R) = R-bar * D3 = 9.7333 *0 = 0
Centreline of R chart is R-bar (R mean) = 9.7333

As it is clearly evident that, all the sample means are lying within the limits as calculated above. Hence the process is in statistical control.

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