Chapter 10, Problem 12P

a)

Summary Introduction

To construct: A $\overline{\text{X}}$ chart with three-sigma limits and check whether the process is controlled or not.

Introduction: Quality is a measure of excellence or a state of being free from deficiencies, defects and important variations. It is obtained by consistent and strict commitment to certain standards to attain uniformity of a product to satisfy consumers’ requirement.

Answer to Problem 12P

The process is not in control.

Explanation of Solution

Given information:

Sample	Mean	Sample	Mean	Sample	Mean	Sample	Mean
1	3.86	11	3.88	21	3.84	31	3.88
2	3.90	12	3.86	22	3.82	32	3.76
3	3.83	13	3.88	23	3.89	33	3.83
4	3.81	14	3.81	24	3.86	34	3.77
5	3.84	15	3.83	25	3.88	x35	3.86
6	3.83	16	3.86	26	3.90	36	3.80
7	3.87	17	3.82	27	3.81	37	3.84
8	3.88	18	3.86	28	3.86	38	3.79
9	3.84	19	3.84	29	3.98	39	3.85
10	3.80	20	3.87	30	3.96

Formula:

$\text{Controllimits}=\overline{\overline{\text{x}}}\pm 3\frac{\sigma }{\sqrt{\text{n}}}$

Calculation of control limits and construction of $\overline{X}$ chart:

${\displaystyle \sum \overline{X}=}150.15$

$\begin{array}{c}\overline{\overline{\text{X}}}=\frac{{\displaystyle \sum _{\text{i}=\text{1}}^{39}\overline{\text{X}}}}{\text{n}}\\ =\frac{150.15}{39}\\ =3.85\end{array}$

$\overline{\overline{\text{X}}}$ is calculated by dividing sum of the means which is 150.18 by 39 which yields 3.85

$\begin{array}{c}\text{Control}\text{limits}=3.85\pm 3\frac{0.146}{\sqrt{14}}\\ =3.85\pm 0.117\\ \text{UCL}=3.967\\ \text{LCL}=3.733\end{array}$

Graph shows the plot for sample means with UCL and LCL values. It can be observed that some points are above the control limits. So, the process is out of control.

Hence, the process is not in control.

b)

Summary Introduction

To analyze: The data using median run test and up and down run test and conclude the results.

Introduction: Quality is a measure of excellence or a state of being free from deficiencies, defects and important variations. It is obtained by consistent and strict commitment to certain standards to attain uniformity of a product to satisfy consumers’ requirement.

Answer to Problem 12P

The results of the median run test and up/down test is random and no non randomness is detected.

Explanation of Solution

Given information:

Sample	Mean	Sample	Mean	Sample	Mean	Sample	Mean
1	3.86	11	3.88	21	3.84	31	3.88
2	3.90	12	3.86	22	3.82	32	3.76
3	3.83	13	3.88	23	3.89	33	3.83
4	3.81	14	3.81	24	3.86	34	3.77
5	3.84	15	3.83	25	3.88	x35	3.86
6	3.83	16	3.86	26	3.90	36	3.80
7	3.87	17	3.82	27	3.81	37	3.84
8	3.88	18	3.86	28	3.86	38	3.79
9	3.84	19	3.84	29	3.98	39	3.85
10	3.80	20	3.87	30	3.96

Formula:

${\text{Z}}_{\text{test}}=\frac{\text{Observed}-\text{Expected}}{\text{Standard}\text{deviation}}$

Analysis of data:

To make analysis of data, the given data is compared with median 3.85 to make A/B and U/D.

Sample	A/B	Mean	U/D	Sample	A/B	Mean	U/D
1	A	3.86	-	21	B	3.84	D
2	A	3.90	U	22	B	3.82	D
3	B	3.83	D	23	A	3.89	U
4	B	3.81	D	24	A	3.86	D
5	B	3.84	U	25	A	3.88	U
6	B	3.83	D	26	A	3.90	U
7	A	3.87	U	27	B	3.81	D
8	A	3.88	U	28	A	3.86	U
9	B	3.84	D	29	A	3.98	U
10	B	3.80	D	30	A	3.96	D
11	A	3.88	U	31	A	3.88	D
12	A	3.86	D	32	B	3.76	D
13	A	3.88	U	33	B	3.83	U
14	B	3.81	D	34	B	3.77	D
15	B	3.83	U	35	A	3.86	U
16	A	3.86	U	36	B	3.80	D
17	B	3.82	D	37	B	3.84	U
18	A	3.86	U	38	B	3.79	D
19	B	3.84	D	39	B	3.85	U
20	A	3.87	U

Sample 39 ties with the median and in order to maximize the Z_test statistics, the sample 39 is labeled as B. Therefore, the observed number of runs is 18.

Median run test:

Calculation of expected number of runs:

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{med}}=\frac{\text{N}}{2}+1\\ =\frac{39}{2}+1\\ =20.5\end{array}$

The expected number of runs is calculated by adding half of the total number of samples with 1 which gives 20.5.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{med}}=\sqrt{\frac{\text{N}-1}{4}}\\ =\sqrt{\frac{39-1}{4}}\\ =3.08\end{array}$

Standard deviation is calculated by subtracting number of sample 39 from 1 and dividing the resultant by 4 and taking square for the value which yields 3.08.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{\text{18}-\text{20}\text{.5}}{\text{3}\text{.08}}\\ =-0.81\end{array}$

The z factor for median is calculated by dividing the difference of 18 and 20.5 with 3.08 which yields -0.81 which is within the test statistics of ±2.

Up/Down Test:

The observed number of runs from the analysis is 29.

Calculation of expected number of runs:

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{u/d}}=\frac{2\text{N}-1}{3}\\ =\frac{\left(2\times 39\right)-1}{3}\\ =25.7\end{array}$

The expected number of runs is calculated by subtracting the double of the number of samples 39 and subtracting from1 and dividing the resultant with 3 which gives 25.7.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{u/d}}=\sqrt{\frac{\text{16N}-29}{90}}\\ =\sqrt{\frac{\left(16\times 39\right)-29}{90}}\\ =2.57\end{array}$

Standard deviation is calculated by multiplying the number of samples with 16 and subtracting the resultant from 29 and then dividing the resulting value with 90 and taking square root which yields 2.57.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{\text{29}-\text{25}\text{.7}}{\text{2}\text{.57}}\\ =+1.28\end{array}$

The z factor for median is calculated by dividing the difference of 29 and 25.7 with 2.57 which yields +1.28 which is within the test statistics of ±2.

Hence, the results of the median run test and up/down test is random and no non randomness is detected.