The Basic Practice of Statistics
The Basic Practice of Statistics
8th Edition
ISBN: 9781319057916
Author: Moore
Publisher: MAC HIGHER
Question
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Chapter 31, Problem 31.61E
To determine

To calculate: The control limits for s when removing sample 2 and 17. Also add the new control limits to the s chart.

To find: The control limits for x¯ when removing the sample 2 and 17. Make the x¯ chart and comment on the control chart.

Expert Solution & Answer
Check Mark

Answer to Problem 31.61E

For the s chart, the center line is 7.73, the lower control limit is 0 and the upper control limit is 19.85.

Output using the MINITAB software is given below:

The Basic Practice of Statistics, Chapter 31, Problem 31.61E , additional homework tip  1

From the s charts, it can be observed that the process is within control.

For the x¯ chart, the center line is 672.95, the lower control limit is 657.8465 and the upper control limit is 688.0535.

Output using the MINITAB software is given below:

The Basic Practice of Statistics, Chapter 31, Problem 31.61E , additional homework tip  2

From the x¯ charts, it can be observed that the process is not within control.

Explanation of Solution

Given info:

The data shows the mean and standard deviation for 22 samples of three films.

Calculation:

s chart:

Center line for an s chart:

CL=s¯=(6.8+10.6+12+5.3+4.5+11+6+11.3+8.3+4.1+3.9+6.1+7.9+8.4+6.2+8.3+11.6+5.7+5.8+10.7)20=154.620=7.73

The value of σ^ is

σ^=s¯c4=7.730.8862(From table 31.3, c4=0.8862)=8.72

Upper control limits:

UCL=B6σ^=2.276(8.72)(From table 31.3,B6(=3)=2.276)=19.85

Lower control limits:

LCL=B5σ^=0(8.72)(From table 31.3,B5(=3)=0)=0

Thus, the center line is 7.73, the lower control limit is 0, and the upper control limit is 19.85.

Software procedure:

Step-by-step procedure to construct s charts using the MINITAB software:

  • Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
  • Choose With Groups and click OK.
  • In Series, select the column of standard deviation.
  • In Show reference lines at Y values, enter 0, 7.73 and 19.85.
  • Click OK.

Observation:

Since all observations lie within the lower and upper control limit, the s chart is within control.

x¯ chart:

Center line for a x¯ chart:

CL=x¯¯=¯13,45920=672.95

The value of σ^ is

σ^=s¯c4=7.730.8862(From table 31.3, c4=0.8862)=8.72

Upper control limits:

UCL=x¯¯+3σ^n=672.95+38.723=672.95+15.1035=688.05

Lower control limits:

LCL=x¯¯3σ^n=672.9538.723=672.9515.1035=657.85

Thus, the center line is 672.95, the lower control limit is 657.85 and the upper control limit is 688.05.

Software procedure:

Step-by-step procedure to construct x¯ charts using the MINITAB software:

  • Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
  • Choose With Groups and click OK.
  • In Series, select the column of mean.
  • In Show reference lines at Y values, enter 672.95, 657.85 and 688.05.
  • Click OK.

Observation:

Since all the observations do not lie within the lower and upper control limit, the x¯ chart is not within control.

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