The Basic Practice of Statistics
The Basic Practice of Statistics
7th Edition
ISBN: 9781464142536
Author: David S. Moore, William I. Notz, Michael A. Fligner
Publisher: W. H. Freeman
Question
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Chapter 31, Problem 31.60E
To determine

The control limits for x¯ chart and describe the control of short term process variation by the construction of x¯ chart after the removal of initial 10 samples.

Expert Solution & Answer
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Answer to Problem 31.60E

The upper control limit and lower control limit for s are 847.136 and 821.864 respectively.

Explanation of Solution

Given info:

The 22 samples of three films each are measured at 10 minutes intervals. In samples 1 to 10, mistakes in operating the interferometer resulted in one high outlier thickness reading that was clearly incorrect.

Calculation:

Since the points for samples 1 and 10 lie above the upper control limit, thus the two values of s are to be removed that were out of control.

Thus the values for rest 20 samples are,

Since the total past samples are 20, thus the formula to calculate the mean is,

x¯¯=number of possible outcomestotal number of outcomes

Thus, the overall mean of the sample means is,

x¯¯=834.5

The overall mean of the standard deviation is,

s¯=6.465

The formula to calculate the process standard deviation is,

σ=s¯c4 (1)

Substitute 6.465 for s¯ and 0.8862 for c4 in equation (1).

σ=6.4650.8862=7.2952

The formula to calculate the center line for s chart is,

μs=c4σ (2)

Substitute 0.8862 for c4 and 7.2952 for σ in equation (2).

CL=(0.8862)(7.2952)=6.465

The formula to calculate the upper control limit for an s chart is,

UCL=B6σ (3)

Substitute 2.276 for B6 and 7.2952 for σ in equation (3).

UCL=(2.276)(7.2952)=16.6038

Thus the UCL is 16.6038.

The formula to calculate the lower control limit for s chart is,

LCL=B5σ (4)

Substitute 0 for B5 and 7.2952 for σ in equation (4).

LCL=(0)(7.2952)=0

Thus the LCL is 0.

Thus the center line and control limits for an s chart are 6.465, 16.6038 and 0 respectively.

The x axis of the s chart shows the sample and y axis shows standard deviation.’

Upper control limit is 16.6038.

Lower control limit is 0.

Centerline of the s chart is 6.465.

The Basic Practice of Statistics, Chapter 31, Problem 31.60E , additional homework tip  1

This shows that all the 20 samples are within the control limits, that is the data is in control.

The formula to calculate the center line for x¯ chart is,

CL=x¯¯ (5)

Substitute 834.5 for x¯¯ in equation (5).

CL=834.5

The formula to calculate the upper control limit for x¯ chart is,

UCL=x¯¯+3(σn) (6)

Substitute 834.5 for x¯¯ , 3 for n and 7.2952 for σ in equation (6).

UCL=834.5+3(7.29523)=834.5+12.6356=847.136

Thus the UCL is 847.136.

The formula to calculate the lower control limit for x¯ chart is,

LCL=x¯¯+3(σn) (7)

Substitute 834.5 for x¯¯ , 3 for n and 7.2952 for σ in equation (7).

LCL=834.53(7.29523)=834.512.6356=821.864

Thus the LCL is 821.864.

Thus the center line and control limits for an s chart are 834.5, 847.136 and 821.864 respectively.

The x axis of the x¯ chart shows the sample and y axis shows mean.

Upper control limit is 847.136.

Lower control limit is 821.864.

Centerline of the s chart is 834.5.

The Basic Practice of Statistics, Chapter 31, Problem 31.60E , additional homework tip  2

This shows that all the 20 samples are within the control limits, which are the data is in control.

Thus, s and x¯ charts are in control.

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