Question

You are considering relaxing your control requirements that determine what is acceptable quality; you have been using a 99.0% confidence interval but want to begin using a 97.5% confidence interval.

Your team has collected the following data from 4 samples of 7 observations each. The calculated standard deviation is 13.981.

	Sample 1	Sample 2	Sample 3	Sample 4
Obs 1	392.2	415.1	413.6	416.2
Obs 2	392.3	408.1	394.9	388.1
Obs 3	405.4	428.6	410.1	408.2
Obs 4	410.3	398.2	410.8	419.5
Obs 5	423.3	403.3	423.2	385.8
Obs 6	413.9	421.1	402.7	431.0
Obs 7	426.7	433.5	385.8	390.2

What is the UCL for the mean given the new confidence interval of 97.5%? (Keep one decimal point in your answer)

Answer 1

Solution:

The Upper control limit (UCL) is calculated as below;

UCL = X-bar + [Z-value x /SQRT(n)]

where,

X-bar = Process mean

Z-value (for 97.5% confidence interval) = 1.96 (Using Excel’s NORMSINV function)

n = Number of observations in each sample = 7

= Standard deviation = 13.981

X-bar is calculated as below:

Sample 1 Average = (392.2 + 392.3 + 405.4 + 410.3 + 423.3 + 413.9 + 426.7) / 7 = 409.2

Sample 2 Average = (415.1 + 408.1 + 428.6 + 398.2 + 403.3 + 421.1 + 433.5) / 7 = 415.4

Sample 3 Average = (413.6 + 394.9 + 410.1 + 410.8 + 423.2 + 402.7 + 385.8) / 7 = 405.9

Sample 4 Average = (415.2 + 409.7 + 405.6 + 423.3 + 385.8 + 431.0 + 390.3) / 7 = 408.7

X-bar = (409.2 + 415.4 + 405.9 + 408.7) / 4

X-bar = 409.8

Putting all the values in the above formula, we get;

UCL = X-bar + [Z-value x /SQRT(n)]

UCL = 409.8 + [1.96 x 13.981/SQRT(7)]

UCL = 409.8 + 10.4

UCL = 420.2