Question

The following are quality control data for a manufacturing process at Kensport Chemical Company. The data show the temperature in degrees centigrade at five points in time during a manufacturing cycle.

Sample	x	R
1	95.72	1.0
2	95.24	0.9
3	95.18	0.9
4	95.46	0.4
5	95.46	0.5
6	95.32	1.1
7	95.40	0.9
8	95.44	0.3
9	95.08	0.2
10	95.50	0.6
11	95.80	0.6
12	95.22	0.2
13	95.56	1.3
14	95.22	0.6
15	95.04	0.8
16	95.72	1.1
17	94.82	0.6
18	95.46	0.5
19	95.60	0.4
20	95.74	0.6

The company is interested in using control charts to monitor the temperature of its manufacturing process. Compute the upper and lower control limits for the R chart. (Round your answers to three decimal places.)

UCL =

LCL =

Construct the R chart.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Range and ranges from 0.00 to 2.00.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at 0.00. The middle horizontal line intersects the vertical axis at about 0.68. The top horizontal line is labeled UCL and intersects the vertical axis at about 1.43. The entire region between the LCL line and the UCL line is shaded.
• All of the points are plotted between the LCL line and the UCL line. The line segments connecting the points don't form any particular shape.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Range and ranges from 0.00 to 2.00.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at about 0.09. The middle horizontal line intersects the vertical axis at about 0.68. The top horizontal line is labeled UCL and intersects the vertical axis at about 1.07. The entire region between the LCL line and the UCL line is shaded.
• Most of the points are plotted between the LCL line and the UCL line. There are 3 points above the UCL line. The line segments connecting the points don't form any particular shape.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Range and ranges from 0.00 to 2.00.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at 0.00. The middle horizontal line intersects the vertical axis at about 0.68. The top horizontal line is labeled UCL and intersects the vertical axis at about 1.28. The entire region between the LCL line and the UCL line is shaded.
• Most of the points are plotted between the LCL line and the UCL line. There is one point above the UCL line. The line segments connecting the points don't form any particular shape.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Range and ranges from 0.00 to 2.00.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at 0.00. The middle horizontal line intersects the vertical axis at about 0.68. The top horizontal line is labeled UCL and intersects the vertical axis at about 1.57. The entire region between the LCL line and the UCL line is shaded.
• All of the points are plotted between the LCL line and the UCL line. The line segments connecting the points don't form any particular shape.

Compute the upper and lower control limits for the

x

chart. (Round your answers to three decimal places.)

UCL = LCL =

Construct the

x

chart.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Mean x bar and ranges from 94.75 to 96.25.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at about 95.01. The middle horizontal line intersects the vertical axis at about 95.40. The top horizontal line is labeled UCL and intersects the vertical axis at about 95.79. The entire region between the LCL line and the UCL line is shaded.
• Most of the points are plotted between the LCL line and the UCL line. There is one point below the LCL line and one point above the UCL line. The line segments connecting the points don't form any particular shape.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Mean x bar and ranges from 94.75 to 96.25.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at about 94.78. The middle horizontal line intersects the vertical axis at about 95.40. The top horizontal line is labeled UCL and intersects the vertical axis at about 96.02. The entire region between the LCL line and the UCL line is shaded.
• All of the points are plotted between the LCL line and the UCL line. The line segments connecting the points don't form any particular shape.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Mean x bar and ranges from 94.75 to 96.25.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at about 94.93. The middle horizontal line intersects the vertical axis at about 95.40. The top horizontal line is labeled UCL and intersects the vertical axis at about 95.87. The entire region between the LCL line and the UCL line is shaded.
• Most of the points are plotted between the LCL line and the UCL line. There is one point below the LCL line. The line segments connecting the points don't form any particular shape.

• A graph shows three horizontal lines and a series of 20 points connected by line segments.
• The horizontal axis is labeled Sample Number and ranges from 0 to 20.
• The vertical axis is labeled Sample Mean x bar and ranges from 94.75 to 96.25.
• The bottom horizontal line is labeled LCL and intersects the vertical axis at about 95.09. The middle horizontal line intersects the vertical axis at about 95.40. The top horizontal line is labeled UCL and intersects the vertical axis at about 95.71. The entire region between the LCL line and the UCL line is shaded.
• Most of the points are plotted between the LCL line and the UCL line. There are 3 points below the LCL line and 4 points above the UCL line. The line segments connecting the points don't form any particular shape.

What conclusions can be made about the quality of the process?

The R chart indicates that the process variability is  ---Select--- in control out of control .  ---Select--- No samples fall One sample falls Two samples fall More than two samples fall outside the R chart control limits. The

x

chart indicates that the process mean is  ---Select--- in control out of control .  ---Select--- No samples fall One sample falls Two samples fall More than two samples fall outside the

x

chart control limits.

Answer 1

Data:

Sample	x	R
1	95.72	1
2	95.24	0.9
3	95.18	0.9
4	95.46	0.4
5	95.46	0.5
6	95.32	1.1
7	95.4	0.9
8	95.44	0.3
9	95.08	0.2
10	95.5	0.6
11	95.8	0.6
12	95.22	0.2
13	95.56	1.3
14	95.22	0.6
15	95.04	0.8
16	95.72	1.1
17	94.82	0.6
18	95.46	0.5
19	95.6	0.4
20	95.74	0.6

no. of samples = 20, size of each sample = 5.

Mean =

95.40

0.68

= 95.40 and = 0.68

Now,

For R chart,
LCL =	D3Rbar
UCL =	D4Rbar

At n = 5, D3 = 0, D4 = 2.114

Then, LCL = 0 and UCL = 0.68*2.114 = 1.438 and CL = 0.68

R chart:

Process is under statistical control (R bar)

Now, x chart,

For X chart,
LCL =	xbar - A2Rbar
UCL =	xbar + A2Rbar

At, n=5, A2 = 0.577

then, LCL = 95.4 - 0.577 * 0.68 = 95

and UCL = 95.4 + 0.577*0.68 = 95.79

and CL = 95.4

then, x chart:

The process is statistically out of control (x bar)

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