Chapter 10, Problem 28P

a)

Summary Introduction

To determine: The normal time for the operation.

Introduction: The 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 28P

The normal time is calculated as 77.8 minutes.

Explanation of Solution

Given information:

The observed time and performance rating for each element is given as follows:

Element	Performance rating	Observation (minutes)
		1	2	3	4	5
Prepare daily reports	120	35	40	33	42	39
Photocopy results	110	12	10	36a	15	13
Label and package reports	90	3	3	5	5	4
Distribute reports	85	15	18	21	17	45b

^a refers to the photocopying machine being broken

^b refers to the delay in allowance factor

Formulae to calculate normal time:

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

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

Calculate normal time:

Element	Performance rating	Observation (minutes)					Average time	Normal time
		1	2	3	4	5
Prepare daily reports	120%	35	40	33	42	39	37.8	45.36
Photocopy results	110%	12	10	36a	15	13	12.5	13.75
Label and package reports	90%	3	3	5	5	4	4	3.6
Distribute reports	85%	15	18	21	17	45b	17.75	15.09

Calculate the average time to prepare daily reports:

The average time is calculated by taking an average of the observation given, which is a summation of 35, 40, 33, 42, and 39 and dividing the result by 5. Hence, the average is 37.8 minutes.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{35+40+33+42+39}{5}\\ =\frac{189}{5}\\ =37.8\text{ minutes}\end{array}$

Calculate the average time to photocopy results:

The average time is calculated by taking an average of the observation given, which is a summation of 12, 10, 15, and 13 and dividing the result by 4. Hence, the average is 12.5 minutes.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{12+10+15+13}{4}\\ =\frac{50}{4}\\ =12.5\text{ minutes}\end{array}$

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

Calculate the average time to label and package reports:

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{3+3+5+5+4}{5}\\ =\frac{20}{5}\\ =4.0\text{ minutes}\end{array}$

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

Calculate the average time to distribute reports:

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{15+18+21+17}{4}\\ =\frac{71}{4}\\ =17.75\text{ minutes}\end{array}$

The average time is calculated by taking an average of the observation given, which is a summation of 15, 18, 21, and 17 and dividing the result by 4. Hence, the average is 17.75 minutes.

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

Calculate the normal time to prepare daily reports:

The normal time is calculated by multiplying average time and performance rating. The average time is calculated as 37.8 minutes, and performance rating is given as 120%. Hence, the normal time is 45.36 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =37.8\times 120\%\\ =37.8\times 1.2\\ =45.36\text{ minutes}\end{array}$

Calculate the normal time to photocopy results:

The normal time is calculated by multiplying average time and performance rating. The average time is calculated as 12.50 minutes, and performance rating is given as 110%. Hence, the normal time is 13.75 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =12.50\times 110\%\\ =12.50\times 1.1\\ =13.75\text{ minutes}\end{array}$

Calculate the normal time to label and package reports:

The normal time is calculated by multiplying average time and performance rating. The average time is calculated as 4.0 minutes, and performance rating is given as 90%. Hence, the normal time is 3.6 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =4.0\times 90\%\\ =4.0\times 0.9\\ =3.6\text{ minutes}\end{array}$

Calculate the normal time to distribute reports:

The normal time is calculated by multiplying average time and performance rating. The average time is calculated as 17.75 minutes, and performance rating is given as 85%. Hence, the normal time is 15.09 minutes.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =17.75\times 85\%\\ =17.75\times 0.85\\ =15.09\text{ minutes}\end{array}$

Calculate the total normal time:

The normal time is calculated by adding the normal time to prepare daily reports, photocopy results, label and package reports, and distribute reports.

$\begin{array}{c}\text{Normal time}=\text{Sum of normal times}\\ =45.36+13.75+3.6+15.09\\ =77.8\text{ minutes}\end{array}$

Hence, the total normal time is 77.8 minutes.

b)

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 28P

Thestandard time is calculated as 91.53 minutes.

Explanation of Solution

Given information:

The allowance is given as 15%. The observed time and performance rating for each element is given as follows:

Element	Performance rating	Observation (minutes)
		1	2	3	4	5
Prepare daily reports	120	35	40	33	42	39
Photocopy results	110	12	10	36a	15	13
Label and package reports	90	3	3	5	5	4
Distribute reports	85	15	18	21	17	45b

^a refers to the photocopying machine being broken

^b refers to the delay in allowance factor

Formula to calculate standard time:

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

Calculate 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 77.8 minutes and allowance factor is calculated as 15%

$\begin{array}{c}\text{Standard time}=\frac{\text{Normal time}}{1-\text{Allowance factor}}\\ =\frac{77.8\text{ minutes}}{1-0.15}\\ =\frac{77.8\text{ minutes}}{0.85}\\ =91.53\text{ minutes}\end{array}$

Hence, standard time is 91.53 minutes.

c)

Summary Introduction

To determine: The number of the sample size to prepare reports.

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

Answer to Problem 28P

The sample size to prepare reports is 15, to photocopy results is 44, label, and package reports is 96, and to distribute reports is 31 samples.

Explanation of Solution

Given information:

The confidence level is 95% and accuracy is ±5%.

Element	Performance rating	Observation (minutes)
		1	2	3	4	5
Prepare daily reports	120	35	40	33	42	39
Photocopy results	110	12	10	36a	15	13
Label and package reports	90	3	3	5	5	4
Distribute reports	85	15	18	21	17	45b

Formula to determine the required number of observation:

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

n refers to the number of observations required

z refers to the table value for the confidence level

s refers to the standard deviation

h refers to an accuracy level

x(bar) refers to the average observed time

Determine a proper number of observation required:

Element	Mean observed time	s	Sample
Prepare daily reports	37.8	3.7	15
Photocopy results	12.5	2.1	44
Label and package reports	4	1	96
Distribute reports	17.75	2.5	31

Calculate the standard deviation to prepare daily reports:

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 results should be square rooted. The standard deviation to prepare the daily report is 3.7.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(35-37.8\right)}^2+{\left(40-37.8\right)}^2+{\left(33-37.8\right)}^2+{\left(42-37.8\right)}^2+{\left(39-37.8\right)}^2}{5-1}}\\ =\sqrt{\frac{54.8}{4}}\\ =3.7\end{array}$

Calculate the standard deviation to photocopy results:

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 results should be square rooted. The standard deviation to photocopy results is 2.1.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{(12-12.5)}^2+{\left(10-12.5\right)}^2+{\left(15-12.5\right)}^2+{\left(13-12.5\right)}^2}{4-1}}\\ =\sqrt{\frac{13}{3}}\\ =2.1\end{array}$

Calculate the standard deviation to label and package reports:

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 results should be square rooted. The standard deviation to label and package reports is 1.

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

Calculate the standard deviation to distribute reports:

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 results should be square rooted. The standard deviation to distribute reports is 2.5.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(15-17.75\right)}^2+{\left(18-17.75\right)}^2+{\left(21-17.75\right)}^2+{\left(17-17.75\right)}^2}{4-1}}\\ =\sqrt{\frac{18.75}{3}}\\ =2.5\end{array}$

Calculate the sample size to prepare daily reports:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 3.7, and dividing the result with the multiple of the accuracy level and average observed time that is given as 0.05 and 37.8, respectively. Hence, the required number of observations is 15 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 3.7}{0.05\times 37.8}\right)}^2\\ ={\left(\frac{7.252}{1.89}\right)}^2\\ =14.72\simeq 15\text{ samples}\end{array}$

Calculate the sample size to photocopy results:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 2.1, and dividing the result with the multiple of the accuracy level and average observed time that is given as 0.05 and 12.5, respectively. Hence, the required number of observations is 44 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 2.1}{0.05\times 12.5}\right)}^2\\ ={\left(\frac{4.116}{0.625}\right)}^2\\ =43.37\simeq 44\text{ samples}\end{array}$

Calculate the sample size to label and package reports:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 1, and dividing the result with the multiple of the accuracy level and average observed time that is given as 0.05 and 4, respectively. Hence, the required number of observations is 96 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 1}{0.05\times 4}\right)}^2\\ ={\left(\frac{1.96}{0.2}\right)}^2\\ =96.04\simeq 96\text{ samples}\end{array}$

Calculate the sample size to distribute reports:

It is calculated by multiplying the z value of 1.96, the standard deviation that is given as 2.5, and dividing the result with the multiple of the accuracy level and average observed time that is given as 0.05 and 17.75, respectively. Hence, the required number of observations is 31 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{1.96\times 2.5}{0.05\times 17.75}\right)}^2\\ ={\left(\frac{4.9}{0.8875}\right)}^2\\ =30.48\simeq 31\text{ samples}\end{array}$

Hence, the normal time for the process is 77.8 minutes and standard time for the process is 91.53 minutes. The sample size to prepare reports is 15, to photocopy results is 44, label and package reports are 96, and to distribute reports is 31 samples.