Chapter 10, Problem 14P

a)

Summary Introduction

To determine: The results of the run test.

Introduction: Run chart refers to a line graph that displays the recorded data based on the time sequence.

Answer to Problem 14P

The result of the run test suggests that the output possess randomness.

Explanation of Solution

Given information:

Test	z-score
Median	+1.37
Up/Down	+1.05

By observing the given data of median +1.37 (< ±2) and up/down +1.05 (< ±2), it can be concluded that they are within the statistical limits of ±2.

Hence, the result of the run test suggests that the output possess randomness.

b)

Summary Introduction

To determine: The results of the run test.

Introduction: Run chart refers to a line graph that displays the recorded data based on the time sequence.

Answer to Problem 14P

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

Explanation of Solution

Given information:

$\begin{array}{c}\text{Number}\text{of}\text{means}=20\\ \text{Number}\text{of}\text{runs}\text{counted}\text{above/below}=14\\ \text{Number}\text{of}\text{runs}\text{counted}\text{up/down}=8\end{array}$

Formula:

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

Calculation of expected number of runs:

Observed number of runs = 14

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{med}}=\frac{\text{N}}{2}+1\\ =\frac{20}{2}+1\\ =11.0\end{array}$

The expected number of runs is calculated by adding half of the total number of samples with 1 which gives 11.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{med}}=\sqrt{\frac{\text{N}-1}{4}}\\ =\sqrt{\frac{20-1}{4}}\\ =2.18\end{array}$

Standard deviation is calculated by subtracting number of sample 20 from 1 and dividing the resultant by 4 and taking square for the value which yields 2.18.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{\text{14}-\text{11}}{\text{2}\text{.18}}\\ =+1.38\end{array}$

The z factor for median is calculated by dividing the difference of 14 and 11 with 2.18 which yields +1.38 which is within the test statistics of ±2.00 and no non-randomness exist.

Up/Down Test:

The observed number of runs is 8.

Calculation of expected number of runs:

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{u/d}}=\frac{2\text{N}-1}{3}\\ =\frac{\left(2\times 20\right)-1}{3}\\ =13.0\end{array}$

The expected number of runs is calculated by subtracting the double of the number of samples 20 and subtracting from1 and dividing the resultant with 3 which gives 13.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{u/d}}=\sqrt{\frac{\text{16N}-29}{90}}\\ =\sqrt{\frac{\left(16\times 20\right)-29}{90}}\\ =1.80\end{array}$

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 1.80.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{8-13.0}{1.80}\\ =-2.78\end{array}$

The z factor for median is calculated by dividing the difference of 8 and 13 with 1.80 which yields -2.78 which is beyond the test statistics of ±2.00 which is non-random.

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

c)

Summary Introduction

To determine: The results of the run test.

Introduction: Run chart refers to a line graph that displays the recorded data based on the time sequence.

Answer to Problem 14P

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

Explanation of Solution

Given information:

Cable	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Defects	2	3	1	0	1	3	2	0	2	1	3	1	2	0

Formula:

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

Analysis of data:

$\begin{array}{c}\overline{\text{c}}=\frac{21}{14}\\ =1.5\end{array}$

To make analysis of data, the given data is compared with median (center line) to make A/B and U/D which is shown below,

Median run test:

From analysis, the observed number of runs is 10 and median is 1.5.

Calculation of expected number of runs:

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{med}}=\frac{\text{N}}{2}+1\\ =\frac{14}{2}+1\\ =8.0\end{array}$

The expected number of runs is calculated by adding half of the total number of samples with 1 which gives 8.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{med}}=\sqrt{\frac{\text{N}-1}{4}}\\ =\sqrt{\frac{14-1}{4}}\\ =1.80\end{array}$

Standard deviation is calculated by subtracting number of sample 14 from 1 and dividing the resultant by 4 and taking square for the value which yields 1.80.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{\text{10}-8}{1.80}\\ =+1.11\end{array}$

The z factor for median is calculated by dividing the difference of 10 and 8 with 1.80 which yields +1.11 which is within the test statistics of ±2.00 and no non-randomness exist.

Up/Down Test:

The observed number of runs from the analysis is 10.

Calculation of expected number of runs:

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{u/d}}=\frac{2\text{N}-1}{3}\\ =\frac{\left(2\times 14\right)-1}{3}\\ =9.0\end{array}$

The expected number of runs is calculated by subtracting the double of the number of samples 14 and subtracting from1 and dividing the resultant with 3 which gives 9.0.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{u/d}}=\sqrt{\frac{\text{16N}-29}{90}}\\ =\sqrt{\frac{\left(16\times 14\right)-29}{90}}\\ =1.47\end{array}$

Standard deviation is calculated by multiplying the number of samples with 14 and subtracting the resultant from 29 and then dividing the resulting value with 90 and taking square root which yields 1.47.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{10-9.0}{\text{1}\text{.47}}\\ =+0.68\end{array}$

The z factor for median is calculated by dividing the difference of 10 and 9 with 1.47 which yields +0.68which is within the test statistics of ±2.00.

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

d)

Summary Introduction

To determine:The results of the run test.

Introduction: Run chart refers to a line graph that displays the recorded data based on the time sequence.

Answer to Problem 14P

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

Explanation of Solution

Given information:

Day	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Comp.	4	10	14	8	9	6	5	12	13	7	6	4	2	10

Formula:

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

Analysis of data:

$\begin{array}{c}\overline{\text{c}}=\frac{110}{14}\\ =7.875\end{array}$

To make analysis of data, the given data is compared with median (center line) to make A/B and U/D which is shown below,

Median run test:

From analysis, the observed number of runs is 8.

Calculation of expected number of runs:

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{med}}=\frac{\text{N}}{2}+1\\ =\frac{14}{2}+1\\ =8.0\end{array}$

The expected number of runs is calculated by adding half of the total number of samples with 1 which gives 8.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{med}}=\sqrt{\frac{\text{N}-1}{4}}\\ =\sqrt{\frac{14-1}{4}}\\ =1.80\end{array}$

Standard deviation is calculated by subtracting number of sample 14 from 1 and dividing the resultant by 4 and taking square for the value which yields 1.80.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{6-8}{1.80}\\ =-1.11\end{array}$

The z factor for median is calculated by dividing the difference of 6 and 8 with 1.80 which yields -1.11 which is beyond the test statistics of ±2.00 and non-randomness exist.

Up/Down Test:

The observed number of runs from the analysis is 7.

Calculation of expected number of runs:

$\begin{array}{c}\text{E}{\left(\text{r}\right)}_{\text{u/d}}=\frac{2\text{N}-1}{3}\\ =\frac{\left(2\times 14\right)-1}{3}\\ =9.0\end{array}$

The expected number of runs is calculated by subtracting the double of the number of samples 26 and subtracting from1 and dividing the resultant with 3 which gives 9.0.

Calculation of standard deviation:

$\begin{array}{c}{\text{σ}}_{\text{u/d}}=\sqrt{\frac{\text{16N}-29}{90}}\\ =\sqrt{\frac{\left(16\times 14\right)-29}{90}}\\ =1.47\end{array}$

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 1.47.

$\begin{array}{c}{\text{Z}}_{\text{test}}=\frac{7-9.0}{\text{1}\text{.47}}\\ =-1.36\end{array}$

The z factor for median is calculated by dividing the difference of 7 and 9 with 1.47 which yields -1.36 which is within the test statistics of ±2.00.

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