EBK STATISTICAL TECHNIQUES IN BUSINESS
EBK STATISTICAL TECHNIQUES IN BUSINESS
17th Edition
ISBN: 9781259924163
Author: Lind
Publisher: MCGRAW HILL BOOK COMPANY
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Chapter 19, Problem 22CE

A new machine has just been installed to produce printed circuit boards. One of the critical measurements is the thickness of the resistance layer. The quality control inspector randomly selects five boards each half-hour, measures the thickness, and records the results. The measurements (in millimeters) for the period 8:00 a.m. to 10:30 a.m. follow.

Chapter 19, Problem 22CE, A new machine has just been installed to produce printed circuit boards. One of the critical

  1. a. Determine the control limits for the mean and the range.
  2. b. Plot the control limits for the mean outside diameter and the range.
  3. c. Are there any points on the mean or the range chart that are out of control? Comment on the chart.

a.

Expert Solution
Check Mark
To determine

Find the control limits for the mean.

Find the control limits for the range.

Answer to Problem 22CE

The control limits for the mean is 86.61, and 88.11.

The control limits for the range is 0, and 2.75.

Explanation of Solution

Calculation:

The mean of the sample means is,

x¯¯=x¯k

Here, x¯ denotes the sum of sample means, and k denotes the number of sample means.

The formula for the control limits is,

LCL,UCL=x¯¯±A2R¯

Here, x¯¯ denotes the mean of the sample means, A2 is the constant value and R¯ denotes the mean of the ranges.

The mean of the sample ranges is,

R¯=Rk

Here, R denotes the sum of sample ranges, and k denotes the number of sample means.

The formula for the control limits is,

LCL,UCL=D3R¯,D4R¯

Here, R¯ denotes the mean of the sample ranges, and D3,D4 are the constant values.

The sample means are ranges are obtained below:

TimeThickness (millimeters)  
12345TotalAverageRange
8:0087.187.387.987.087.0436.387.26(=436.35)0.9(=87.987.0)
8:3086.988.587.687.587.4437.987.58(=437.95)1.6(=88.586.9)
9:0087.588.486.987.688.2438.687.72(=438.65)1.5(=88.486.9)
9:3086.088.087.287.687.1435.987.18(=435.95)2(=88.086.0)
10:0087.187.187.187.187.1435.587.1(=435.55)0(=87.187.1)
10:3088.086.287.487.387.8436.787.34(=436.75)1.8(=88.086.2)
 Totals524.187.8

The mean of the sample means is,

x¯¯=524.186=87.36

Hence, the mean of the sample means is 87.36.

The mean of the sample ranges is,

R¯=7.86=1.3

Hence, the mean of the sample ranges it as 1.3.

Compute the value of A2 by using factors for control chart.

From the Appendix B: Tables B.10 Factors for Control Charts:

  • Locate the value A2 in chart for Averages row.
  • Locate the value 5 in n column.
  • The intersecting value is 0.577.

The control limits for the mean is,

LCL,UCL=87.36±(0.577×1.3)=87.36±0.75=(87.360.75,87.36+0.75)=86.61,88.11

Hence, the control limits for the mean is 86.61, and 88.11.

Compute the values of D3 and D4 by using factors for control chart.

From the Appendix B: Tables B.10 Factors for Control Charts:

  • Locate the value D3 in chart for Ranges row.
  • Locate the value 5 in n column.
  • The intersecting value is 0.
  • Locate the value D4 in chart for Ranges row.
  • Locate the value 5 in n column.
  • The intersecting value is 2.115.

The control limits for the range is,

LCL,UCL=0(1.3),(2.115)(1.3)=0,2.75

Hence, the control limits for the range is 0, and 2.75.

b.

Expert Solution
Check Mark
To determine

Construct the control limits for the mean.

Construct the control limits for the range.

Answer to Problem 22CE

Output using MINITAB software is given below:

EBK STATISTICAL TECHNIQUES IN BUSINESS, Chapter 19, Problem 22CE , additional homework tip  1

EBK STATISTICAL TECHNIQUES IN BUSINESS, Chapter 19, Problem 22CE , additional homework tip  2

Explanation of Solution

Calculation:

Step by step procedure to obtain x¯ chart using MINITAB software is given as,

  • Choose Stat > Control Charts > Variables charts for subgroups > Xbar Chart.
  • Select option Observations for a subgroup are in one row of columns.
  • Enter the column Thickness 1, Thickness 2, Thickness 3, Thickness 4, and Thickness 5.
  • Choose Xbar Options.
  • In Estimate, select Rbar under Method for estimating standard deviation.
  • Click OK.

Step by step procedure to obtain R chart using MINITAB software is given as,

  • Choose Stat > Control Charts > Variables charts for subgroups > R Chart.
  • Select option Observations for a subgroup are in one row of columns.
  • Enter the column Thickness 1, Thickness 2, Thickness 3, Thickness 4, and Thickness 5.
  • Choose R Options.
  • In Estimate, select Rbar under Method for estimating standard deviation.
  • Click OK.

c.

Expert Solution
Check Mark
To determine

Check whether there any points on the mean or the range chart that are out of control.

Answer to Problem 22CE

The points are within the control.

Explanation of Solution

Calculation:

For x¯ chart:

The limits are 86.61 to 88.11 any value beyond this limits is considered as outlier.

In the chart no values are outside the limits this indicates that the process is in the control which implies that none of values are out of control.

For R chart:

The limits are 0 to 2.75 any value beyond this limits is considered as outlier.

In the chart no values are outside the limits this indicates that the process is in the control which implies that none of values are out of control.

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Chapter 19 Solutions

EBK STATISTICAL TECHNIQUES IN BUSINESS

Ch. 19 - Auto-Lite Company manufactures car batteries. At...Ch. 19 - Below is a p-chart for a manufacturing process. a....Ch. 19 - Inter-State Moving and Storage Company is setting...Ch. 19 - A bicycle manufacturer randomly selects 10 frames...Ch. 19 - During the process of producing toilet paper,...Ch. 19 - Sams Supermarkets monitors the checkout scanners...Ch. 19 - Dave Christi runs a car wash chain with outlets...Ch. 19 - Compute the probability of accepting a lot of DVDs...Ch. 19 - Determine the probability of accepting lots that...Ch. 19 - Determine the probability of accepting lots that...Ch. 19 - Warren Electric manufactures fuses for many...Ch. 19 - Grills Video Products purchases LCDs from Mira...Ch. 19 - The production supervisor at Westburg Electric...Ch. 19 - The manufacturer of running shoes conducted a...Ch. 19 - At Rumseys Old Fashion Roast Beef, cola drinks are...Ch. 19 - A new machine has just been installed to produce...Ch. 19 - Long Last Tire Company, as part of its inspection...Ch. 19 - Charter National Bank has a staff of loan officers...Ch. 19 - Prob. 25CECh. 19 - Early Morning Delivery Service guarantees delivery...Ch. 19 - An automatic machine produces 5.0-millimeter bolts...Ch. 19 - Steele Breakfast Foods Inc. produces a popular...Ch. 19 - An investor believes there is a 5050 chance that a...Ch. 19 - Lahey Motors specializes in selling cars to buyers...Ch. 19 - A process engineer is considering two sampling...Ch. 19 - Christina Sanders is a member of the womens...Ch. 19 - Erics Cookie House sells chocolate chip cookies in...Ch. 19 - The numbers of near misses recorded for the last...Ch. 19 - Prob. 35CECh. 19 - Swiss Watches, Ltd. purchases watch stems for...Ch. 19 - Automatic Screen Door Manufacturing Company...Ch. 19 - Prob. 38CE
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