Applied Statistics Exercise 9.3

Dear Class 12 Samacheer Kalvi students, here are the Text Book Solutions for Exercise 9.3 Business Maths Chapter 9 Applied Statistics for your reference and study.

You can find the links to the other exercises below.

Applied Statistics Important Formula

Solutions for Applied Statistics Exercise 9.1

Solutions for Applied Statistics Exercise 9.2

Solutions for Applied Statistics Exercise 9.4

Applied Statistics Exercise 9.3

$\text{1. Define Statistical Quality Control.}\\$
$\text{Statistical Quality Control (SQC) is the application of statistical techniques}\\$
$\text{to monitor, control, and improve the quality of products or processes. It uses}\\$
$\text{tools such as control charts, sampling inspection, and process capability analysis}\\$
$\text{to ensure that the production process operates at the desired level of quality.} \\$

$\text{2. Mention the types of causes for variation in a production process.}\\$
$\text{There are two causes of variation which affects the quality of a product:} \\$
$\text{1. Chance Causes (or) Random causes}\\$
$\text{2. Assignable Causes}\\$

$\text{3. Define Chance Cause.} \\$
$\text{These are small variations which are natural and inherent in the manufacturing} \\$
$\text{process. The variation occurring due to these causes is beyond the human control}\\$
$\text{and cannot be prevented or eliminated under any circumstances. The minor causes} \\$
$\text{which do not affect the quality of the products to an extent are called as Chance }\\$
$\text{Causes (or) Random causes. For example Rain, floods, power cuts, etc.} \\$

$\text{4. Define Assignable Cause.}\\$
$\text{The variation which is present in any production process is due to non-random}\\$
$\text{causes is called Assignable causes. It may occur in at any stage of the process,} \\$
$\text{right from the arrival of the raw materials to the final delivery of the product.} \\$
$\text{Some of the important factors of assignable causes are defective raw materials,} \\$
$\text{fault in machines, unskilled manpower, worn out tools, new operation, etc.} \\$

$\text{5. What do you mean by product control?}\\$
$\text{Product Control means that controlling the quality of the product} \\$
$\text{by critical examination through sampling inspection plans. Product} \\$
$\text{Control aims at a certain quality level to be guaranteed to the customers.} \\$
$\text{It attempts to ensure that the product sold does not contain a large number} \\$
$\text{of defective items. Thus it is concerned with classification of raw materials,} \\$
$\text{semi-finished goods or finished goods into acceptable or rejectable products.} \\$

$\text{6. What do you mean by process control?}\\$
$\text{The main objective in any production process is to control and maintain}\\$
$\text{a satisfactory quality level of the manufactured product. This is done} \\$
$\text{by 'Process Control'. In Process Control the proportion of defective}\\$
$\text{ items in the production process is to be minimized and it is achieved}\\$
$\text{through the technique of control charts.} \\$

$\text{7. Define a control chart.} \\$
$\text{A control chart is a statistical tool used to study how a process}\\$
$\text{ changes over time by plotting sample data against predetermined}\\$
$\text{ control limits, helping to distinguish between normal process }\\$
$\text{variation and variation caused by assignable factors.} \\$

$\text{8. Name the control charts for variables.}\\$
$\text{The control charts for variables include:} .}\\$
$\text{(i) Charts for Mean} ( \bar {X} ) .}\\$
$\text{(ii) Charts for Range (R)} .}\\$

$\text{9. Define mean chart.}\\$
$\text{A mean chart is a type of control chart used to monitor the average or }\\$
$\text{mean value of a quality characteristic of a process over time, helping}\\$
$\text{to determine whether the process mean is in a state of statistical control.} \\$

$\text{10. Define R Chart.}\\$
$\text{An R-chart is a control chart used to monitor the variability or dispersion} \\$
$\text{ of a process by plotting the ranges of sample subgroups over time, helping}\\$
$\text{ to determine whether the process variation is in statistical control.}\\$

$\text{11. What are the uses of statistical quality control?} \\$
$\text{The main purpose of SQC is to device statistical techniques which would help in}\\$
$\text{elimination of assignable causes and bring the production process under control.}\\$

$\text{12. Write the control limits for the mean chart.} \\$

$\text{When }\bar {X} \text{and SD are given}\\$

$UCL=\bar {\bar X}+3\frac{\sigma}{\sqrt{n}}\\$

$CL=\bar {\bar X} \\$

$LCL=\bar {\bar X}-3\frac{\sigma}{\sqrt{n}} \\$

$\text{When }\bar {X} \text{and SD are not given}\\$

$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$CL=\bar {\bar X} \\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$

$\text{13. Write the control limits for the R chart.} \\$

$\text{case (i) when SD is given}\\$

$UCL=\bar { R}+3\sigma_{R}\\$
$CL=\bar { R}\\$
$LCL=\bar { R}-3\sigma_{R}\\$

$\text{case (i) when SD is not given}\\$

$UCL=D_{4}\bar { R}\\$
$CL=\bar { R}\\$
$LCL=D_{3}\bar { R}\\$

$\text{14. A machine is set to deliver packets of a given weight.Ten samples of size}\\$
$\text{ five each were recorded. Below are given relevant data:}\\$

Sample number	1	2	3	4	5	6	7	8	9	10
X̄	15	17	15	18	17	14	18	15	17	16
R	7	7	4	9	8	7	12	4	11	5

$\text{Calculate the control limits for mean chart and the range chart and then}\\$
$\text{ comment on the state of control. (conversion factors for n = 5,} \; A_{2} =\\$
$0.58,\; D_{3} = 0 \; \text{and }\; D_{4} = 2.115 )\\$

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{No. of samples}\\$
$\bar {\bar X}==\dfrac{162}{10}\\$
$\bar {\bar X}=16.2\\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{74}{10}\\$
$\bar {R}=7.4\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=16.2+(0.58)(7.4) \\$
$UCL=16.2+4.292\\$
$UCL=20.492\\$
$CL=\bar {\bar X}=16.2\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=16.2-(0.58)(7.4) \\$
$LCL=16.2-4.292\\$
$LCL=11.908\\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.115)(7.4) \\$
$UCL=15.651\\$
$CL=\bar {R}=7.4\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(7.4) \\$
$LCL=0\\$

$\text{Conclusion: Since all the points in the mean chart and the range chart}\\$
$\text{ fall within the control limits, the process is in control.} \\$

$\text{15. Ten samples each of size five are drawn at regular intervals} \\$
$\text{from a manufacturing process. The sample means}(\bar{X})\text{and their} \\$
$\text{ranges (R) are given below:} \\$

Sample number	1	2	3	4	5	6	7	8	9	10
X̄	49	45	48	53	39	47	46	39	51	45
R	7	5	7	9	5	8	8	6	7	6

$\text{Calculate the control limits in respect of X chart. ( Given } \\$
$A_{2} = 0.58\; D_{3} = 0 \;\text{ and}\; D_{4} = 2.115. \text{ Comment on the state of control.}\\$

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{\text{No. of samples}}\\$
$\bar {\bar X}==\dfrac{462}{10}\\$
$\bar {\bar X}=46.2\\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{68}{10}\\$
$\bar {R}=6.8\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=46.2+(0.58)(6.8) \\$
$UCL=46.2+3.944\\$
$UCL=50.144\\$
$CL=\bar {\bar X}=46.2\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=46.2-(0.58)(6.8) \\$
$LCL=46.2-3.944\\$
$LCL=42.256\\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.115)(6.8) \\$
$UCL=14.382\\$
$CL=\bar {R}=6.8\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(6.8) \\$
$LCL=0\\$

$\text{Conclusion: Since all the points in the mean chart falls} \\$
$\text{ outside the control limits, the process is out of control.} \\$

$\text{16. Construct }\;\bar{X}\; \text{and R charts for the following data:}\\$

Sample	Number	Observations
1	32	36	42
2	28	32	40
3	39	52	28
4	50	42	31
5	42	45	34
6	50	29	21
7	44	52	35
8	22	35	44

$\text{(Given for n = 3,}\; A_{2} = 0.58,\; D_{3} = 0 \;\text{and}\; D_{4} = 2.115 )\\$

Sample Number	x̄	R
1	36.7	10
2	33.3	12
3	39.7	24
4	41	19
5	40.3	11
6	33.3	29
7	43.7	17
8	33.7	22

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{\text{No. of samples}}\\$
$\bar {\bar X}==\dfrac{301.7}{8}\\$
$\bar {\bar X}=37.71\\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{144}{8}\\$
$\bar {R}=18\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=37.71+(1.023)(18) \\$
$UCL=37.71+18.414\\$
$UCL=56.124\\$
$CL=\bar {\bar X}=37.71\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=37.71-(1.023)(18) \\$
$LCL=37.71-18.414\\$
$LCL=19.296\\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.574)(18) \\$
$UCL=46.332\\$
$CL=\bar {R}=18\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(18) \\$
$LCL=0\\$

$\text{Conclusion: Since all the points in the mean chart and the range chart}\\$
$\text{ fall within the control limits, the process is in control.} \\$

$\text{17. The following data show the values of sample mean}\; \bar(X)\; \text{and its}\\$
$\text{ range (R ) for the samples of size five each. Calculate the values for control}\\$
$\text{ limits for mean , range chart and determine whether the process is in control.}\\$

Sample Number	Mean	Range
1	11.2	7
2	11.8	4
3	10.8	8
4	11.6	5
5	11	7
6	9.6	4
7	10.4	8
8	9.6	4
9	10.6	7
10	10	9

$\text{( conversion factors for n }\;= 5, A_{2} = 0.58 ,D_{3} = 0 \;\text{and }\;D_{4} = 2.115 )\\$

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{\text{No. of samples}}\\$
$\bar {\bar X}==\dfrac{106.6}{10}\\$
$\bar {\bar X}=10.66\\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{63}{10}\\$
$\bar {R}=6.3\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=10.66+(0.58)(6.3) \\$
$UCL=10.66+3.654\\$
$UCL=14.314\\$
$CL=\bar {\bar X}=10.66\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=10.66-(0.58)(6.3) \\$
$LCL=10.66-3.654\\$
$LCL=7.006\\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.115)(6.3) \\$
$UCL=13.3245\\$
$CL=\bar {R}=6.3\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(6.3) \\$
$LCL=0\\$

$\text{Conclusion: Since all the points in the mean chart and the range chart}\\$
$\text{ fall within the control limits, the process is in control.} \\$

$\text{18. A quality control inspector has taken ten samples of size four packets}\\$
$\text{ each from a potato chips company.The contents of the sample are given }\\$
$\text{below, Calculate the control limits for mean and range chart.}\\$

Sample Number	Observations
	1	2	3	4
1	12.5	12.3	12.6	12.7
2	12.8	12.4	12.4	12.8
3	12.1	12.6	12.5	12.4
4	12.2	12.6	12.5	12.3
5	12.4	12.5	12.5	12.5
6	12.3	12.4	12.6	12.6
7	12.6	12.7	12.5	12.8
8	12.4	12.3	12.6	12.5
9	12.6	12.5	12.3	12.6
10	12.1	12.7	12.5	12.8

Sample Number	x̄	R
1	12.5	0.4
2	12.6	0.4
3	12.4	0.5
4	12.4	0.4
5	12.5	0.1
6	12.5	0.3
7	12.7	0.3
8	12.5	0.3
9	12.5	0.3
10	12.5	0.7

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{\text{No. of samples}}\\$
$\bar {\bar X}==\dfrac{125.1}{10}\\$
$\bar {\bar X}=12.51\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{37}{10}\\$
$\bar {R}=3.7\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=12.51+(0.58)(0.37) \\$
$UCL=12.51+0.2146\\$
$UCL=12.72\\$
$CL=\bar {\bar X}=12.51\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=12.51-(0.58)(0.37) \\$
$LCL=12.51-0.2146\\$
$LCL=12.30\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.115)(0.37) \\$
$UCL=0.78\\$
$CL=\bar {R}=0.37\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(0.37) \\$
$LCL=0\\$

$\text{Conclusion: Since all the points in the mean chart and the range chart}\\$
$\text{ fall within the control limits, the process is in control.} \\$

$\text{19. The following data show the values of sample means and the ranges for ten}\\$
$\text{ samples of size 4 each. Construct the control chart for mean and range chart }\\$
$\text{and determine whether the process is in control.}\\$

Sample Number	x̄	R
1	29	39
2	26	10
3	37	39
4	34	17
5	14	12
6	45	20
7	39	5
8	20	21
9	34	23
10	23	15

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{\text{No. of samples}}\\$
$\bar {\bar X}==\dfrac{301}{10}\\$
$\bar {\bar X}=30.1\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{201}{10}\\$
$\bar {R}=20.1\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=30.1+(0.729)(20.1) \\$
$UCL=30.1+14.73\\$
$UCL=44.93\\$
$CL=\bar {\bar X}=30.1\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=30.1-(0.729)(20.1) \\$
$LCL=30.1-14.73\\$
$LCL=15.37\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.282)(20.1) \\$
$UCL=45.87\\$
$CL=\bar {R}=20.1\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(20.1) \\$
$LCL=0\\$

$\text{Conclusion: Since two points in the mean chart fall outside}\\$
$\text{ the control limits, the process is out of control.} \\$

$\text{20. In a production process, eight samples of size 4 are collected and their means }\\$
$\text{and ranges are given below. Construct mean chart and range chart with control limits.}\\$

Sample Number	x̄	R
1	12	2
2	13	5
3	11	4
4	12	2
5	14	3
6	13	2
7	16	4
8	15	3

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{\text{No. of samples}}\\$
$\bar {\bar X}==\dfrac{106}{8}\\$
$\bar {\bar X}=13.25\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{25}{8}\\$
$\bar {R}=3.125\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=13.25+(0.729)(3.125) \\$
$UCL=13.25+2.28\\$
$UCL=15.53\\$
$CL=\bar {\bar X}=13.25\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=13.25-(0.729)(3.125) \\$
$LCL=30.1-2.28\\$
$LCL=10.97\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.282)(3.125) \\$
$UCL=7.14\\$
$CL=\bar {R}=3.125\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(3.125) \\$
$LCL=0\\$

$\text{Conclusion: Since one point in the mean chart lies outside}\\$
$\text{ the control limits, the process is out of control.} \\$

$\text{21. In a certain bottling industry the quality control inspector recorded} \\$
$\text{the weight of each of the 5 bottles selected at random during each hour} \\$
$\text{ of four hours in the morning.}\\$

Time	Weights in ml
8:00 AM	43	41	42	43	41
9:00 AM	40	39	40	39	44
10:00 AM	42	42	43	38	40
11:00 AM	39	43	40	39	42

$\bar {\bar X}=\dfrac{\Sigma \bar{X}}{\text{No. of samples}}\\$
$\bar {\bar X}==\dfrac{164}{4}\\$
$\bar {\bar X}=41\$
$\bar {R}=\dfrac{\Sigma R}{n}\\$
$\bar {R}=\dfrac{16}{4}\\$
$\bar {R}=4\\$

$\text{The control limits for the mean chart are:}\\$
$UCL=\bar {\bar X}+A_{2}\bar {R}\\$
$UCL=41+(0.577)(4) \\$
$UCL=41+2.308\\$
$UCL=43.31\\$
$CL=\bar {\bar X}=41\\$
$LCL=\bar {\bar X}-A_{2}\bar {R}\\$
$LCL=41-(0.577)(4) \\$
$LCL=41-2.308\\$
$LCL=38.69\$

$\text{The control limits of the range chart are:}\\$
$UCL=D_{4}\bar {R}\\$
$UCL=(2.114)(4) \\$
$UCL=8.46\\$
$CL=\bar {R}=4\\$
$LCL=D_{3}\bar {R}\\$
$LCL=0(4) \\$
$LCL=0\\$

$\text{Conclusion: Since all the points in the mean chart and the range chart}\\$
$\text{ fall within the control limits, the process is in control.} \\$

Applied Statistics Exercise 9.3

Applied Statistics Exercise 9.3

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Applied Statistics Exercise 9.3

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