Samples of size 5 provided the following 20 sample means for a production process that is believed to be in control. 95.72 95.24 95.18 95.44 95.46 95.32 95.40 95.44 95.08 95.50 95.80 95.22 95.56 95.22 95.04 95.72 94.82 95.46 95.60 95.78 a. Based on these data, what is an estimate of the mean when the process is in control? b. Assume that the process standard deviation is σ = .50. Develop the x ¯ control chart for this production process. Assume that the mean of the process is the estimate developed in part (a). c. Do any of the 20 sample means indicate that the process was out of control?

Question

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Answer 1

Textbook Question

Chapter 19, Problem 16SE

Samples of size 5 provided the following 20 sample means for a production process that is believed to be in control.

95.72	95.24	95.18
95.44	95.46	95.32
95.40	95.44	95.08
95.50	95.80	95.22
95.56	95.22	95.04
95.72	94.82	95.46
95.60	95.78

a. Based on these data, what is an estimate of the mean when the process is in control?
b. Assume that the process standard deviation is σ = .50. Develop the x ¯ control chart for this production process. Assume that the mean of the process is the estimate developed in part (a).
c. Do any of the 20 sample means indicate that the process was out of control?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

a.

Expert Solution

To determine

Find the estimate of the mean when the process is in control.

Answer to Problem 16SE

The estimate of the mean for production process that is in control is 95.4.

Explanation of Solution

Calculation:

The 20 sample means for a production process are provided.

The estimate of the mean for production process is given below:

μ=∑x¯n=(95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46)20=1,90820=95.4

Thus, the estimate of the mean for production process that is in control is 95.4.

b.

Expert Solution

To determine

Construct a x¯ control chart for the production process.

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Statistics for Business & Economics, Chapter 19, Problem 16SE

Explanation of Solution

Calculation:

Upper control limits for x¯chart:

UCL=μ+3σn=95.4+30.505=95.4+0.6696=96.07

Lower control limits for x¯chart:

LCL=μ−3σn=95.4−30.505=95.4−0.6696=94.73

Software procedure:

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

Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
Choose Simple, and then click OK.
In Series, select Mean.
In Show reference lines at Y values, enter 96.07, 95.4, and 94.73.
Click OK.

Thus, the control limits for x¯ chart are UCL=96.07 and LCL=94.73.

c.

Expert Solution

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Explanation of Solution

From the x¯ chart, it is observed that all the sample means lie within the UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.

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Answer 2

Textbook Question

Chapter 19, Problem 16SE

Samples of size 5 provided the following 20 sample means for a production process that is believed to be in control.

95.72	95.24	95.18
95.44	95.46	95.32
95.40	95.44	95.08
95.50	95.80	95.22
95.56	95.22	95.04
95.72	94.82	95.46
95.60	95.78

a. Based on these data, what is an estimate of the mean when the process is in control?
b. Assume that the process standard deviation is σ = .50. Develop the x ¯ control chart for this production process. Assume that the mean of the process is the estimate developed in part (a).
c. Do any of the 20 sample means indicate that the process was out of control?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

a.

Expert Solution

To determine

Find the estimate of the mean when the process is in control.

Answer to Problem 16SE

The estimate of the mean for production process that is in control is 95.4.

Explanation of Solution

Calculation:

The 20 sample means for a production process are provided.

The estimate of the mean for production process is given below:

μ=∑x¯n=(95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46)20=1,90820=95.4

Thus, the estimate of the mean for production process that is in control is 95.4.

b.

Expert Solution

To determine

Construct a x¯ control chart for the production process.

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Statistics for Business & Economics, Chapter 19, Problem 16SE

Explanation of Solution

Calculation:

Upper control limits for x¯chart:

UCL=μ+3σn=95.4+30.505=95.4+0.6696=96.07

Lower control limits for x¯chart:

LCL=μ−3σn=95.4−30.505=95.4−0.6696=94.73

Software procedure:

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

Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
Choose Simple, and then click OK.
In Series, select Mean.
In Show reference lines at Y values, enter 96.07, 95.4, and 94.73.
Click OK.

Thus, the control limits for x¯ chart are UCL=96.07 and LCL=94.73.

c.

Expert Solution

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Explanation of Solution

From the x¯ chart, it is observed that all the sample means lie within the UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Textbook Question

Chapter 19, Problem 16SE

Samples of size 5 provided the following 20 sample means for a production process that is believed to be in control.

95.72	95.24	95.18
95.44	95.46	95.32
95.40	95.44	95.08
95.50	95.80	95.22
95.56	95.22	95.04
95.72	94.82	95.46
95.60	95.78

a. Based on these data, what is an estimate of the mean when the process is in control?
b. Assume that the process standard deviation is σ = .50. Develop the x ¯ control chart for this production process. Assume that the mean of the process is the estimate developed in part (a).
c. Do any of the 20 sample means indicate that the process was out of control?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

a.

Expert Solution

To determine

Find the estimate of the mean when the process is in control.

Answer to Problem 16SE

The estimate of the mean for production process that is in control is 95.4.

Explanation of Solution

Calculation:

The 20 sample means for a production process are provided.

The estimate of the mean for production process is given below:

μ=∑x¯n=(95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46)20=1,90820=95.4

Thus, the estimate of the mean for production process that is in control is 95.4.

b.

Expert Solution

To determine

Construct a x¯ control chart for the production process.

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Statistics for Business & Economics, Chapter 19, Problem 16SE

Explanation of Solution

Calculation:

Upper control limits for x¯chart:

UCL=μ+3σn=95.4+30.505=95.4+0.6696=96.07

Lower control limits for x¯chart:

LCL=μ−3σn=95.4−30.505=95.4−0.6696=94.73

Software procedure:

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

Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
Choose Simple, and then click OK.
In Series, select Mean.
In Show reference lines at Y values, enter 96.07, 95.4, and 94.73.
Click OK.

Thus, the control limits for x¯ chart are UCL=96.07 and LCL=94.73.

c.

Expert Solution

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Explanation of Solution

From the x¯ chart, it is observed that all the sample means lie within the UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

Chapter 19, Problem 16SE

Samples of size 5 provided the following 20 sample means for a production process that is believed to be in control.

95.72	95.24	95.18
95.44	95.46	95.32
95.40	95.44	95.08
95.50	95.80	95.22
95.56	95.22	95.04
95.72	94.82	95.46
95.60	95.78

a. Based on these data, what is an estimate of the mean when the process is in control?
b. Assume that the process standard deviation is σ = .50. Develop the x ¯ control chart for this production process. Assume that the mean of the process is the estimate developed in part (a).
c. Do any of the 20 sample means indicate that the process was out of control?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

a.

Expert Solution

To determine

Find the estimate of the mean when the process is in control.

Answer to Problem 16SE

The estimate of the mean for production process that is in control is 95.4.

Explanation of Solution

Calculation:

The 20 sample means for a production process are provided.

The estimate of the mean for production process is given below:

μ=∑x¯n=(95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46)20=1,90820=95.4

Thus, the estimate of the mean for production process that is in control is 95.4.

b.

Expert Solution

To determine

Construct a x¯ control chart for the production process.

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Statistics for Business & Economics, Chapter 19, Problem 16SE

Explanation of Solution

Calculation:

Upper control limits for x¯chart:

UCL=μ+3σn=95.4+30.505=95.4+0.6696=96.07

Lower control limits for x¯chart:

LCL=μ−3σn=95.4−30.505=95.4−0.6696=94.73

Software procedure:

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

Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
Choose Simple, and then click OK.
In Series, select Mean.
In Show reference lines at Y values, enter 96.07, 95.4, and 94.73.
Click OK.

Thus, the control limits for x¯ chart are UCL=96.07 and LCL=94.73.

c.

Expert Solution

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Explanation of Solution

From the x¯ chart, it is observed that all the sample means lie within the UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

a.

Expert Solution

To determine

Find the estimate of the mean when the process is in control.

Answer to Problem 16SE

The estimate of the mean for production process that is in control is 95.4.

Explanation of Solution

Calculation:

The 20 sample means for a production process are provided.

The estimate of the mean for production process is given below:

μ=∑x¯n=(95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46)20=1,90820=95.4

Thus, the estimate of the mean for production process that is in control is 95.4.

b.

Expert Solution

To determine

Construct a x¯ control chart for the production process.

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Statistics for Business & Economics, Chapter 19, Problem 16SE

Explanation of Solution

Calculation:

Upper control limits for x¯chart:

UCL=μ+3σn=95.4+30.505=95.4+0.6696=96.07

Lower control limits for x¯chart:

LCL=μ−3σn=95.4−30.505=95.4−0.6696=94.73

Software procedure:

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

Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
Choose Simple, and then click OK.
In Series, select Mean.
In Show reference lines at Y values, enter 96.07, 95.4, and 94.73.
Click OK.

Thus, the control limits for x¯ chart are UCL=96.07 and LCL=94.73.

c.

Expert Solution

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Explanation of Solution

From the x¯ chart, it is observed that all the sample means lie within the UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.

Answer 8

a.

Expert Solution

To determine

Find the estimate of the mean when the process is in control.

Answer to Problem 16SE

The estimate of the mean for production process that is in control is 95.4.

Explanation of Solution

Calculation:

The 20 sample means for a production process are provided.

The estimate of the mean for production process is given below:

μ=∑x¯n=(95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46)20=1,90820=95.4

Thus, the estimate of the mean for production process that is in control is 95.4.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

Find the estimate of the mean when the process is in control.

Answer 13

Answer to Problem 16SE

The estimate of the mean for production process that is in control is 95.4.

Answer 14

Explanation of Solution

Calculation:

The 20 sample means for a production process are provided.

The estimate of the mean for production process is given below:

μ=∑x¯n=(95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46)20=1,90820=95.4

Thus, the estimate of the mean for production process that is in control is 95.4.

Answer 15

b.

Expert Solution

To determine

Construct a x¯ control chart for the production process.

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Statistics for Business & Economics, Chapter 19, Problem 16SE

Explanation of Solution

Calculation:

Upper control limits for x¯chart:

UCL=μ+3σn=95.4+30.505=95.4+0.6696=96.07

Lower control limits for x¯chart:

LCL=μ−3σn=95.4−30.505=95.4−0.6696=94.73

Software procedure:

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

Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
Choose Simple, and then click OK.
In Series, select Mean.
In Show reference lines at Y values, enter 96.07, 95.4, and 94.73.
Click OK.

Thus, the control limits for x¯ chart are UCL=96.07 and LCL=94.73.

Answer 16

b.

Expert Solution

Answer 17

b.

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

Construct a x¯ control chart for the production process.

Answer 20

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Statistics for Business & Economics, Chapter 19, Problem 16SE

Answer 21

Explanation of Solution

Calculation:

Upper control limits for x¯chart:

UCL=μ+3σn=95.4+30.505=95.4+0.6696=96.07

Lower control limits for x¯chart:

LCL=μ−3σn=95.4−30.505=95.4−0.6696=94.73

Software procedure:

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

Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
Choose Simple, and then click OK.
In Series, select Mean.
In Show reference lines at Y values, enter 96.07, 95.4, and 94.73.
Click OK.

Thus, the control limits for x¯ chart are UCL=96.07 and LCL=94.73.

Answer 22

c.

Expert Solution

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Explanation of Solution

From the x¯ chart, it is observed that all the sample means lie within the UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.

Answer 23

c.

Expert Solution

Answer 24

c.

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Answer 27

Explanation of Solution

From the x¯ chart, it is observed that all the sample means lie within the UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.

Answer 28

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Answer to Problem 16SE

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Answer to Problem 16SE

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Chapter 19 Solutions