Question

To determine if their 14⁢oz filling machine is properly adjusted, Grover Soft Drinks has decided to use an x‾-Chart which uses the range to estimate the variability in the sample.

Period   obs1   obs2   obs3   obs4   obs5   Sample Mean Sample Range
1   13.95   13.98   13.98   13.96   13.97   13.9680 0.03
2   14.02   13.98   13.98   14.00   13.97 13.9900 0.05
3   13.96   13.95   14.01   13.97   13.97   13.9720 0.06
4   13.98   13.99   14.03   13.97   14.01   13.9960 0.06
5   14.04   13.96   14.00   14.02   14.03   14.0100 0.08
6   13.97   14.04   13.99   14.00   14.02   14.0040 0.07
7   13.96   14.00   14.00   13.99   14.02   13.9940 0.06
8   13.97   14.03   13.98   13.99   14.02   13.9980 0.06
9   14.01   14.02   13.96   13.97   14.01   13.9940 0.06
10   14.04   14.04   13.99   14.04   13.99   14.0200 0.05
11   14.03   14.01   14.03   14.02   14.04   14.0260 0.03
12   14.02   13.97   13.97   14.00   13.98   13.9880 0.05
13   13.99   13.98   13.99   14.00   13.99   13.9900 0.02

Step 1 of 7: What is the Center Line of the control chart? Round your answer to three decimal places.

Step 2 of 7: What is the Upper Control Limit? Round your answer to three decimal places.

Step 3 of 7: What is the Lower Control Limit? Round your answer to three decimal places.

Step 4 of 7: Use the following sample data, taken from the next time period, to determine if the process is "In Control" or "Out of Control". Observations: 14.05,14.04,14.05,13.96,14.0814.05,14.04,14.05,13.96,14.08 Sample Mean: 14.036

Step 5 of 7: Use the following sample data, taken from the next time period, to determine if the process is "In Control" or "Out of Control". Observations: 13.95,13.97,13.95,14.04,14.0113.95,13.97,13.95,14.04,14.01 Sample Mean: 13.984

Step 6 of 7: Use the following sample data, taken from the next time period, to determine if the process is "In Control" or "Out of Control". Observations: 14.02,14.05,14.03,14.02,14.0714.02,14.05,14.03,14.02,14.07 Sample Mean: 14.038

Step 7 of 7: Based on the control limits established, what is the probability that the production manager will conclude that the process is "Out of Control", when the process is actually "In Control"? Round your answer to three decimal places.

Answer 1

The data values are,

Period	Obs 1	Obs 2	Obs 3	Obs 4	Obs 5	Sample Mean	Sample Range
1	13.95	13.98	13.98	13.96	13.97	13.968	0.03
2	14.02	13.98	13.98	14	13.97	13.99	0.05
3	13.96	13.95	14.01	13.97	13.97	13.972	0.06
4	13.98	13.99	14.03	13.97	14.01	13.996	0.06
5	14.04	13.96	14	14.02	14.03	14.01	0.08
6	13.97	14.04	13.99	14	14.02	14.004	0.07
7	13.96	14	14	13.99	14.02	13.994	0.06
8	13.97	14.03	13.98	13.99	14.02	13.998	0.06
9	14.01	14.02	13.96	13.97	14.01	13.994	0.06
10	14.04	14.04	13.99	14.04	13.99	14.02	0.05
11	14.03	14.01	14.03	14.02	14.04	14.026	0.03
12	14.02	13.97	13.97	14	13.98	13.988	0.05
13	13.99	13.98	13.99	14	13.99	13.99	0.02
					Sum	181.95	0.68

Step 1 of 7: What is the Center Line of the control chart? Round your answer to three decimal places.

The central line of the control chart is,

Step 2 of 7: What is the Upper Control Limit? Round your answer to three decimal places.

Where, the value of is obtained from control chart factor table,

For sample size = 5 ,

Step 3 of 7:

Step 4 of 7:

Since the sample mean = 14.036 is not within control limit (13.966,14.023), it is "Out of Control"

Step 5 of 7:

Since the sample mean = 13.984 is within control limit (13.966,14.023), it is "In Control"

Step 6 of 7:

Since the sample mean = 14.038 is not within control limit (13.966,14.023), it is "Out of Control"

Step 7 of 7:

The probability that the production manager will conclude that the process is "Out of Control", when the process is actually "In Control" is known as the type I error. The probability is obtained as follow,

Let the sample mean values are normally distributed, the required probability is obtained by applying central limit theorem,

From the data values,

Mean	13.99615
Std.Dev	0.016441
UCL	14.02634
LCL	13.96597

Using the standard normal distribution table for z = -1.8357 and z = 1.8357, the probabilities are

The probability of type I error = 0.066