The Johnson transformation appears to do a superior job to the Box-Cox approach in transforming the data into a normal distribution. As indicated in part b the observed percentage of product out is 24% which is less than the 34% predicted using the normal assumption.

a. Construct a histogram of the data and comment on its normality
b. Construct X and MR charts and comment on the state of statistical control.
c. Using the data and control charts from part b, estimate the mean and standard deviation of the service times.
d. Estimate the percentage of services that are likely to exceed the stipulated time limit.
e. If the process variation cannot be reduced, where would the process have to be centered so that there will be no more that 3.4 failures per million opportunities?
f. Assuming that the average service time remains at the level computed in part c, by what percentage would the process variance have to be reduced so that no more than 3.4 failures per million opportunities are produced?

Observation Measured Observation Measured Observation Measured 22.7098 28.5818 22.3013 20.2175 17.6546 16.3910 21.3819 25.6197 21.2647 23.6092 20.0449 27.705O 21.0183 22.6822 24.1459 30.5566 25.1578 17.3187 16.0317 19.3129 21 16 5825 24 1377 22 8709 22 0960 214009 23.5266 23. 8764 26.3259 26. 7252 23. 8997 24 1986 22. 3498 19.9131 21 3818 20. 9749 31 3345 23.4646 20.9570 18 0233 24. 3035 41 42 43 20.6712 20.2990 24.0512 22.1972 20.1181 21.8725 23.5723 22.2233 23.3868 23.2332 23.5161 26.1952 30.6933 24.8193 19.0917 23.0411 23.8348 28.5909 22.3928 17.3403 23 24 25 26 27 28 29 30 31 32 45 46 47 48 49 50 51 52 53 54 10 12 13 14 15 16 17 18 19 20 34 35 36 37 38 39 40 56 57 59