Chapter 10, Problem 30P

a)

Summary Introduction

To determine: The standard time for the operation.

Introduction: Time study is the basis that helps set the standard time; it would time a sample of the performance of the worker.

Answer to Problem 30P

Thestandard time is calculated as 85.56 minutes.

Explanation of Solution

Given information:

Fatigue factor is given as 10% and personal-time factor is 10%. Accuracy rate is ±5%. The observed time and performance rating for each element is given as follows:

Element	Performance rating	Observation (minutes)
		1	2	3	4	5	6	7
Select correct mufflers	110%	4	5	4	6	4	15a	4
Remove old muffler	90%	6	8	7	6	7	6	7
Weld/install new muffler	105%	15	14	15	12	15	16	13
Check/inspect work	100%	3	4	24a	5	4	3	18a
Complete paperwork	130%	5	6	8	-	7	6	7

^a refers to the fact that the employee has lengthy conversations with the boss (not job-related)

Formulae to calculate standard time:

$\text{Average time}=\frac{\text{Observations}}{n}$

$\text{Normal time}=\text{Average time}\times \text{Performance rating}$

$\text{Standard time}=\frac{\text{Normal time}}{1-\text{Allowance factor}}$

Calculate normal time:

Element	Performance rating	Observation (minutes)							Average time	Normal time
		1	2	3	4	5	6	7
Select correct mufflers	110%	4	5	4	6	4	15a	4	4.50	4.95
Remove old muffler	90%	6	8	7	6	7	6	7	6.71	6.04
Weld/install new muffler	105%	15	14	14	12	15	16	13	14.14	14.85
Check/inspect work	100%	3	4	24a	5	4	3	18a	3.80	3.80
Complete paperwork	130%	5	6	8	-	7	6	7	6.50	8.45

Excel worksheet:

Calculate the average time to select correct mufflers:

The average time is calculated by taking an average of the observation given, which is the summation of 4, 5, 4, 6, 4, and 4 and dividing the result by 6. Hence, the average is 4.5 minutes.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{4+5+4+6+4+4}{6}\\ =\frac{27}{6}\\ =4.50\text{ minutes}\end{array}$

Note: The observation with (X^a) should not be considered while calculating the average time.

Calculate the average time to remove the old muffler:

The average time is calculated by taking an average of the observation given, which is the summation of 6, 8, 7, 6, 7, 6, and 7 and dividing the result by 7. Hence, the average is 6.71 minutes.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{6+8+7+6+7+6+7}{7}\\ =\frac{47}{7}\\ =6.71\text{ minutes}\end{array}$

Calculate the average time to weld or install a new muffler:

The average time is calculated by taking an average of the observation given, which is the summation of 15, 14, 14, 12, 15, 16, and 13 and dividing the result by 7. Hence, the average is 14.14 minutes.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{15+14+14+12+15+16+13}{7}\\ =\frac{99}{7}\\ =14.14\text{ minutes}\end{array}$

Calculate the average time to check or inspect work:

The average time is calculated by taking an average of the observation given, which is the summation of 3, 4, 5, 4, and 3 and dividing the result by 5. Hence, the average is 3.80 minutes.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{3+4+5+4+3}{5}\\ =\frac{19}{5}\\ =3.80\text{ minutes}\end{array}$

Note: The observation with (X^a) should not be considered while calculating the average time.

Calculate the average time to complete paperwork:

The average time is calculated by taking an average of the observation given, which is the summation of 5, 6, 8, 7, 6, and 7 and dividing the result by 6. Hence, the average is 6.50 minutes.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{5+6+8+7+6+7}{6}\\ =\frac{39}{6}\\ =6.50\text{ minutes}\end{array}$

Calculate the normal time to select correct mufflers:

The normal time is calculated by multiplying the average time and performance rating. Average time is calculated as 4.50 minutes and performance rating is given as 110%. Hence, the normal time is 4.95 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =4.50\times 110\%\\ =4.50\times 1.1\\ =4.95\text{ minutes}\end{array}$

Calculate the normal time to remove the old muffler:

The normal time is calculated by multiplying the average time and performance rating. Average time is calculated as 6.71 minutes and performance rating is given as 90%. Hence, the normal time is 6.04 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =6.71\times 90\%\\ =6.71\times 0.90\\ =6.04\text{ minutes}\end{array}$

Calculate the normal time to weld or install a new muffler:

The normal time is calculated by multiplying the average time and performance rating. Average time is calculated as 14.14 minutes and performance rating is given as 105%. Hence, the normal time is 14.85 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =14.14\times 105\%\\ =14.14\times 1.05\\ =14.85\text{ minutes}\end{array}$

Calculate the normal time to check or inspect work:

The normal time is calculated by multiplying the average time and performance rating. Average time is calculated as 3.80 minutes and performance rating is given as 100%. Hence, the normal time is 3.80 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =3.80\times 100\%\\ =3.80\times 1\\ =3.80\text{ minutes}\end{array}$

Calculate the normal time to complete paperwork:

The normal time is calculated by multiplying the average time and performance rating. Average time is calculated as 6.50 minutes and performance rating is given as 130%. Hence, the normal time is 8.45 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =6.50\times 130\%\\ =6.50\times 1.3\\ =8.45\text{ minutes}\end{array}$

Calculate the total normal time:

The normal time is calculated by multiplying the average time and performance rating. Average time is calculated as 6.50 minutes and performance rating is given as 130%. Hence, the normal time is 8.45 minutes.

$\begin{array}{c}\text{Normal time}=\text{Sum of normal times}\\ =4.95+6.04+14.85+3.80+8.45\\ =38.09\text{ minutes}\end{array}$

Normal time is calculated by adding the normal time to select correct mufflers, remove the old muffler, weld or install a new muffler, check or inspect work, and to complete paperwork.

Hence, the total normal time is 38.09 minutes.

Calculate the standard time:

The standard time is calculated by dividing the normal time with the value attained by subtracting the allowance factor from 1. Normal time is calculated as 38.09 minutes and allowance factor is calculated as the sum of fatigue factor, which is 10% and personal time factor, which is 10%.

$\begin{array}{c}\text{Standard time}=\frac{\text{Normal time}}{1-\text{Allowance factor}}\\ =\frac{\text{38}\text{.09 minutes}}{1-\left(0.10+0.10\right)}\\ =\frac{\text{38}\text{.09 minutes}}{1-0.20}\\ =47.6\text{ minutes}\end{array}$

Hence, the standard time is 47.6 minutes.

b)

Summary Introduction

To determine: The number of sample size.

Introduction: Thesample size is required to determine the number of observations that are necessary to find the true cycle time.

Answer to Problem 30P

Thesample size to select correct mufflers is 53, to remove the old muffler is 20, to weld or install a new muffler is 14, to check or inspect work is 75, and to complete paperwork is 40 samples.

Explanation of Solution

Given information:

Confidence level is 95% and accuracy is ±5%.

Element	Performance rating	Observation (minutes)
		1	2	3	4	5	6	7
Select correct mufflers	110%	4	5	4	6	4	15a	4
Remove old muffler	90%	6	8	7	6	7	6	7
Weld/install new muffler	105%	15	14	15	12	15	16	13
Check/inspect work	100%	3	4	24a	5	4	3	18a
Complete paperwork	130%	5	6	8	-	7	6	7

^a refers to the fact that the employee has lengthy conversation with the boss (not job-related)

Formulae to determine the required number of observation:

$n={\left(\frac{zs}{h\overline{x}}\right)}^2$

n refers to number of observations required

z refers to the table value for confidence level

s refers to standard deviation

h refers to accuracy level

$\overline{x}$ refers to average observed time

Determine the proper number of observations required:

Element	Performance rating	Observation (minutes)							Average time	s	Sample
		1	2	3	4	5	6	7
Select correct mufflers	110%	4	5	4	6	4	15a	4	4.50	0.836	53.1
Remove old muffler	90%	6	8	7	6	7	6	7	6.71	0.755	19.5
Weld/install new muffler	105%	15	14	14	12	15	16	13	14.14	1.345	13.9
Check/inspect work	100%	3	4	24a	5	4	3	18a	3.80	0.836	74.5
Complete paperwork	130%	5	6	8	-	7	6	7	6.50	1.048	40.0

Excel worksheet:

Calculate the standard deviation to select correct mufflers:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the result should be square rooted. Standard deviation to select correct mufflers is 0.836.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(4-4.5\right)}^2+{\left(5-4.5\right)}^2+{\left(4-4.5\right)}^2+{\left(6-4.5\right)}^2+{\left(4-4.5\right)}^2+{\left(4-4.5\right)}^2}{6-1}}\\ =\sqrt{\frac{0.25+0.25+0.25+2.25+0.25+0.25}{5}}\end{array}$

$\begin{array}{c}=\sqrt{\frac{3.5}{5}}\\ =\sqrt{0.7}\\ =0.836\end{array}$

Calculate standard deviation to remove old muffler:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the result should be square rooted. Standard deviation to remove old muffler is 0.755.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(6-6.71\right)}^2+{\left(8-6.71\right)}^2+{\left(7-6.71\right)}^2+{\left(6-6.71\right)}^2+{\left(7-6.71\right)}^2+{\left(6-6.71\right)}^2+{\left(7-6.71\right)}^2}{7-1}}\\ =\sqrt{\frac{0.5041+1.6641+0.0841+0.5041+0.0841+0.5041+0.0841}{6}}\end{array}$

$\begin{array}{c}=\sqrt{\frac{3.4287}{6}}\\ =\sqrt{0.57145}\\ =0.755\end{array}$

Calculate standard deviation to weld or install new muffler:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the result should be square rooted. Standard deviation to weld or install new muffler is 1.345.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{\left(\begin{array}{l}{\left(15-14.14\right)}^2+{\left(14-14.14\right)}^2+{\left(14-14.14\right)}^2+{\left(12-14.14\right)}^2+\\ {\left(15-14.14\right)}^2+{\left(16-14.14\right)}^2+{\left(13-14.14\right)}^2\end{array}\right)}{7-1}}\\ =\sqrt{\frac{0.7396+0.0196+0.0196+4.5796+0.7396+3.4596+1.2996}{6}}\end{array}$

$\begin{array}{c}=\sqrt{\frac{10.8572}{6}}\\ =\sqrt{1.809}\\ =1.345\end{array}$

Calculate standard deviation to check or inspect work:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the result should be square rooted. Standard deviation to check or inspect work is 0.836.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(3-3.80\right)}^2+{\left(4-3.80\right)}^2+{\left(5-3.80\right)}^2+{\left(4-3.80\right)}^2+{\left(3-3.80\right)}^2}{5-1}}\\ =\sqrt{\frac{0.64+0.04+1.44+0.04+0.64}{4}}\end{array}$

$\begin{array}{c}=\sqrt{\frac{2.8}{4}}\\ =\sqrt{0.7}\\ =0.836\end{array}$

Calculate standard deviation to complete paperwork:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the result should be square rooted. Standard deviation to check or inspect work is 1.048.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(5-6.5\right)}^2+{\left(6-6.5\right)}^2+{\left(8-6.5\right)}^2+{\left(7-6.5\right)}^2+{\left(6-6.5\right)}^2+{\left(7-6.5\right)}^2}{6-1}}\\ =\sqrt{\frac{2.25+0.25+2.25+0.25+0.25+0.25}{5}}\end{array}$

$\begin{array}{c}=\sqrt{\frac{5.5}{5}}\\ =\sqrt{1.1}\\ =1.048\end{array}$

Calculate the sample size to select correct mufflers:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 0.836, and dividing the resultant with the multiple of the accuracy level and the average observed time that is given as 0.05 and 4.5, respectively. Hence, the required number of observations is 53 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 0.836}{0.05\times 4.5}\right)}^2\\ ={\left(\frac{1.63856}{0.225}\right)}^2\\ \simeq 53\text{ samples}\end{array}$

Calculate the sample size to remove the old muffler:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 0.755, and dividing the resultant with the multiple of the accuracy level and the average observed time that is given as 0.05 and 6.71, respectively. Hence, the required number of observations is 20 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 0.755}{0.05\times 6.71}\right)}^2\\ ={\left(\frac{1.4798}{0.3355}\right)}^2\\ \simeq 20\text{ samples}\end{array}$

Calculate the sample size to weld or install a new muffler:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 1.345, and dividing the resultant with the multiple of the accuracy level and the average observed time that is given as 0.05 and 14.14, respectively. Hence, the required number of observations is 14 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 1.345}{0.05\times 14.14}\right)}^2\\ ={\left(\frac{2.6362}{0.707}\right)}^2\\ \simeq 14\text{ samples}\end{array}$

Calculate the sample size to check or inspect work:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 0.836, and dividing the resultant with the multiple of the accuracy level and the average observed time that is given as 0.05 and 3.8, respectively. Hence, the required number of observations is 75 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 0.836}{0.05\times 3.8}\right)}^2\\ ={\left(\frac{1.63856}{0.19}\right)}^2\\ \simeq 75\text{ samples}\end{array}$

Calculate the sample size to complete paperwork:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 1.048, and dividing the resultant with the multiple of the accuracy level and the average observed time that is given as 0.05 and 6.5, respectively. Hence, the required number of observations is 40 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 1.048}{0.05\times 6.5}\right)}^2\\ ={\left(\frac{2.054}{0.325}\right)}^2\\ \simeq 40\text{ samples}\end{array}$

Hence, the standard time for the process is 47.6 minutes. The sample size to select correct mufflers are 53, to remove the old muffler is 20, weld or install a new muffler is 14, to check or inspect work is 75 samples, and to complete paperwork is 40 samples.