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Lecture- 8

(Quality Management)

Estimation of Population Mean, Sample Size and

Analysis of Variance

Dr. Ali Nesar Khan

Adjunct Faculty

8.1 EstimationE s t im a t i o n is the calculated approximation of a result which is usable

when input data are incomplete/uncertain or extremely difficult to

gather/procure. In mathematics, approximation or estimation typically

means to find upper and/or lower bounds of quantity or database that

cannot readily be computed readily.

Estimation of Population Mean from a Sample Mean

When a sample mean is used as point estimate of population some error

can be expected. Calculating the error, one can find the highest and

lowest values of the mean of the population. In that case,

Estimated population mean = Confidence interval for unknown

mean and unknown standard deviation = Sample mean margin of

error = Sample mean Limit of confidence level

Margin of error/limit of confidence level, m = Se. t =(tn

)

Where, Se =Standard error & t is upper critical value for t

distribution with degree of freedom n-1.

Estimate of population mean, = x m = x tn

=x (value of SE X value of t)

Example-4: Say 12 pieces of dyed fabrics were selected randomly from

dyeing-printing finishing mills in different times within a shift to

determine the dimensional stability (say shrinkage). The unit was

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coloring knit fabrics using winch machines. The test results are as

follows:

4.12% 4.60% 4.40% 4.50% 4.83% 5.00% 5.10% 4.17% 4.51% 4.52% 4.59% 4.22%

From these test results, estimate the mean shrinkage value of the all thefabrics processed in that shift.

x = 4.55

= 0.34

n

E

S = 0.34/12 =0.34/3.16=0.098 (in %)

This value of 0.098 is to be treated with t values for degree of freedom

12-1 = 11.

For 5% probability of error i.e., 100-5 =95% confidence limit, the value of

t against 11 is determined to be 2.201 from t table (as shown in

annexure).

Limit of confidence interval = value of SE XValue of t = 2.201*0.098 = 0.22

The mean of the population

= Sample mean Limit of confidence interval

= x (value of SE XValue of t)

= 4.55 0.22

In other words, the estimated mean of the population will lie within the

range 4.33to 4.77.

Example-5: Say 10 bobbins were tested from 5 ring frames (each with

506 spindles) to determine the lea strengths (in Ibs) and the results wereas shown below:

61, 57.5, 59, 62, 58.5, 62, 59, 57, 63, 62.5

From these test results, estimate the mean strength and standarddeviation of the population (i.e., of 2,530 bobbins in ring frame)

x = 60.15 Ibs

= 2.2

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n

ES = 2.2/3.16 = 0.70 Ibs

This value of 0.70 is to be treated with t values for degree of freedom 10-

1 = 9.

For 5% probability of error i.e., 100-5 =95% confidence limit, the value of

t against 9 is determined to be 2.26 from t table (as shown in annexure).

The mean of the population

= Sample mean (value of SEx XValue of t)

=60.15 (2.26*0.7) = 60.15 1.58

In other words, the mean of the population will lie within the range58.57Ibs to 61.73 Ibs.

Now 95% confidence limit or 5% probability of error corresponds to t

value 1.96. In that case, the value of confidence interval for standard

deviation = SE*1.96 = 0.70*1.96 = 1.73.

The estimated value of standard deviation of the population

= SE*1.96 =2.2 1.73

i.e., standard deviation of the population will range from 0.47 to 3.93 .

8.2 Difference between the Means of Two Large Samples

Example-6: 2 Units A & B are spinning same count of yarn using same

machines and same cotton following same spinning plan. The test results

are shown below:

Unit A Unit B

Mean Strength 60 Ibs 64 Ibs

Standard deviation 4 Ibs 5 IbsSample Tested 100 80

Estimate difference in mean count between these 2 samples?

The estimate of standard deviation of the two samples

SD ( )2

)1()1(

21

2

22

211

nn

nn

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280100

5)180(4)1100( 22

280100

5*794*9922

= 4.94

The sample is large- together forming 180.

At 95% confidence interval i.e., the value of population mean is given by

x 1 -x 2 1.96 ((1/n1+1/n2) = 64-60 1.96*4.94 (1/100+1/80) =40.22

The interval is between 3.78 to 4.22 Ibs.

The difference of mean strength between A & B mills is 3.78 to 4.22 Ibs

at 95% confidence limit.

178

4351

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8.3 Sample Size & Number of Tests to be conducted

The most frequently asked question concerning sampling in the industry

and industrial research is, "What should be the sample size and how

many test one should carry out?" The answer to this question is

influenced by a number of factors, including the purpose of the study,

population size, the risk of selecting a "bad" sample, and the allowable

sampling error. In addition to the purpose of the study and population

size, three criteria usually will need to be specified to determine the

appropriate sample size: t h e l e ve l o f p re c i s i o n / a c cu ra c y o f t e s t

r e s u l t s , t h e l evel o f con f i d ence or r i s k , and thedegr e e of va r i a b i l i t y

i n t h e a t t r i bu t es being measured or tested.

A probability of 5% or 95% confidence is mostly used in textile testing,

QC activities and researches. ASTM recommends determining the sample

size with the help of the following formula,

N = (1.96/E)2

Where, E is error level

is the standard deviation

Example-7: A knitting factory with all its 25 knit machines of same

specification (same gauge and diameter) was producing cotton fabric of

same construction using 100% cotton yarn of Ne 28. In order to check

proper function of knitting machines, a gramage test (weight in gram per

square meter) was conducted by taking 5 meters of fabrics from 5 knit

machines of with the following results:

161g, 190g,174g, 188g, 193g

Was the selection of sample correct?

Mean Count, x 1 = 181.2

Standard Deviation, = 13.44

Standard error,n

ES =13.44/5 =13.44/2.24 =6.0

The value of t at 95% confidence level = 2.77

Population mean (confidence interval) = 181.22.77*6 = 181.2 16.62,

Now this limit of error (or limit of confidence) i.e., 16.62 corresponds to

100*16.62/181.2=9.17% of the mean.

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In textile testing the error is not allowed over 5%. So let us first calculate

this value of error limit or confidence limit for 5%, which corresponds to

error level =5*16.62 /9.17 = 9.06So, at 5% probability error the value of E= 9.06

Number of sample or sample size,

N = (1.96* /E)2

= (1.96*13.44/9.06)2

=8.46

Therefore, the number of samples to be tested is 9.

8.4 Use ofVariances for Significance Test

In order to test significance between 2 sets of variables by variance, the

followings are to be done:

determine variances of each population of variables;

calculate variance ratio with the formula (F = Variance expected

to be greater/Variance Expected to be lower)

compare value of variance ratio (also called F-Ratio) with v1/v2

table specified for corresponding significance limit; Evaluate significance of difference and acceptability level in terms

of significance level.

Example-8: Two jigger machines (Say A & B) in an woven dyeing unit

were coloring same grey fabrics at identical process condition (same

recipes and same shed condition). It was observed for a week whether

they were working with same performance or not. Samples (say 25 from

A and 31 from B) were tested to determine shed percent. The test results

showed standard deviations of 7 percent and 5 percent respectively for A

& B with a slight variation in mean shade percent (say 2% to 5%) amongthem. Now how can one be sure about the equal performance of those 2

machines in terms of shade development?

We are to calculate first the variance ratio of the samples such as:

Variance A = (SDA)2 = 2 =72 = 49,

Variance B= (SDB)2 = 2 52 = 25

Variance or F Ratio,

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F = Variance expected to be greater/Variance Expected to be lower

=49/25 = 1.96

Now this value has to be consulted with Table V1/V2

V1 = 25-1 =24, V2 = 31-1=30

At 5% significance level of probability error, V1=24 & V2 =30, the value of

F ratio is 1.89

At 1% significance level at V1=24 & V2 =30, the value of F ratio is 2.47

2.47>1.96>1.89

So, the difference of work of machines A & B is significant at 5%

significance level.

8.5 Analysis of Variance

The analysis of variance is a technique for splitting up the total variation

observed in the data into different components as per classification of the

data.

Example-8: Four looms A,B,C and D were used to produce a certain kind

of cotton fabric. Four (4) samples from each looms (say by 100 square

meters) were selected from the outputs of each looms at random, and the

number of flaws found in each of 100 square meters were as shownbelow:

A B C D

8 6 14 20

9 8 12 22

11 10 18 25

12 4 9 23

I s t h e r e a si g n i f i c a n t d i f f e r en c e i n t h e p er f o rma n c e o f t h e 4

mach i n e s ?

If there is no significant difference in the performance in the performanceof the 4 machines i.e.

Ho:1= 2= 3= 4

This called null hypothesis.

A B C D

8 6 14 20

9 8 12 22

11 10 18 25

12 4 9 23

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A =40 B=28 C=53 D=90

Ai 2 =82+92+112+122

=410

Bi2

=62+82+102+42=216

Ci2=

142+122+182+92=745

Di2=

202+222+252+232=2038

Grand Total G = A+B+C+D =40+28+53+90 =211

Raw Sum of Square (RSS) = Ai 2+Bi2+Ci2+Di2= 410+216+745+2038

= 3409

Correction Factor (CF) = Square of grand total/Frequency = G2/n =

(211)2/16

=2782.56

Sum of Square of total (SST) = RSS-CF = 3409-2782.56 =626.44

Treatment with Sum of Square (TSS)= Average of Sum of Square-CF

= (A 2+B2+C2+D2)/4-2782.56

= (402+282+532+902)/4 -2782.56

= 13293/4-2782.56 =540.69

Error with sum of square (ESS) = SST-TSS = 626.44-540.59 =85.85

ANOVA Table

Source ofVariation

Sum ofSquare

Degree ofFreedom(DF)

Ratio (MSS)= SS/DF

Variance Ratio=MSSt/MSSe

Table Value

Treatment 540.69 K =4-1=3 180.23 F=180.23/7.15= 25.21

5.95 (at 1%)& 3.49 (at5%)Obtainedfrom table2.11a b

Error 85.85 N =16-4 =12 7.15

25.21>5.95>3.49The table value for F(2.11b) at 1% level of significance is 5.95. The

calculated value of variance ratio is 25.21 which is greater than the table

value; this also rejects the null hypothesis and concludes that there is

significant difference in the performance of the 4 machines at 1%

significance level of F.

The value for variance ratio F (2.11a) at 5% level of significance is 3.49.

The calculated value of variance ratio is 25.21, which is rather greater

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than the table value; this rejects the null hypothesis and concludes that

there is significant difference in the performance of the 4 machines at 5%

significance level of F.Table-1: Significance limit of t

Degrees of freedom Probability

5 per cent 1 per cent123456789101112131415161718192021222324252627282930

406012000

12.7064.3033.1822.7762.5712.4472.3652.3062.2622.2282.2012.1792.1602.1452.1312.1202.1102.1012.0932.0862.0802.0742.0692.0642.0602.0562.0522.0482.0452.042

2.0212.0001.9801.960

63.6579.9255.8414.6044.0323.7073.4993.3553.2503.1693.1063.0553.0122.9772.9472.9212.8982.8782.8612.8452.8312.8192.8072.7972.872.7792.7712.7632.7562.750

2.7042.6602.6172.576

(Abridged from Table III of Statistical Tables of Biological, Agricultural, and Medical Resarch (R.A.Fisher and F.

Yates) (Oliuer and Boyd, Edinburgh). By courtesy of the Authors and Publishers.)

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Table-2: Variance or F ratios (5 percent significance limits of F)

V1

v2

1 2 3 4 5 6 8 12 24 00

12345

678910

11

12131415

1617181920

21222324

25

2627282930

406012000

161.418.5110.137.176.61

5.995.595.325.124.96

4.84

4.754.674.604.54

4.494.454.414.384.35

4.324.304.284.26

4.24

4.224.214.204.184.17

4.084.003.923.84

199.519.009.556.945.79

5.144.744.464.264.10

3.98

3.883.803.743.68

3.633.593.553.523.49

3.473.443.423.40

3.38

3.373.353.343.333.32

3.233.153.072.99

215.719.169.286.595.41

4.764.354.073.863 .71

3.59

3.493.413.343.29

3.243.203.163.133.10

3.073.053.033.01

2.99

2.982.962.952.932.92

2.842.762.682.60

224.619.269.126.395.19

4.534.123.843.633.48

3.36

3.263.183.113.06

3.012.962.932.902.87

2.842.822.802.78

2.76

2.742.732.712.702.69

2.612.522.452.37

230.219.309.016.265.05

4.393.973.693.483.33

3.20

3.113.022.962.90

2.852.812.772.742.71

2.682.662.642.62

2.60

2.592.572.562.542.53

2.452.372.292.21

234.019.338.946.164.95

4.283.873.583.373.22

3.09

3.002.922.852.79

2.742.702.662.632.60

2.572.552.532.51

2.49

2.472.462.442.432.42

2.342.252.172.09

238.919.378.846.044.82

4.153.733.443.233.07

2.95

2.852.772.702.64

2.592.552.512.482.45

2.422.402.382.36

2.34

2.322.302.292.282.27

2.182.102.021.94

243.919.418.745.914.68

4.003.573.283.072.91

2.79

2.692.602.532.48

2.422.382.342.312.28

2.252.232.202.18

2.16

2.152.132.122.102.09

2.001.921.831.75

249.019.458.645.774.53

3.843.413.122.902.74

2.61

2.502.422.352.29

2.242.192.152.112.08

2.052.032.001.98

1.96

1.951.931.911.901.89

1.791.701.611.52

254.319.508.835.634.36

3.673.232.932.712.54

2.40

2.302.212.132.07

2.011.961.921.881.84

1.811.781.761.73

1.71

1.691.671.651.641.62

1.511.391.251.00

(Abridged from Table III of Statistical Tables of Biological, Agricultural, and Medical Resarch (R.A.Fisher and F.

Yates) (Oliuer and Boyd, Edinburgh).

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Table-3: Variance or F ratios (1percent significance limits of F)

V2/v1 1 2 3 4 5 6 8 12 24 00

12345

678910

1112

131415

1617181920

2122232425

2627282930

406012000

405298.4934.1221.2016.26

13.7412.2511.2610.5610.04

9.659.33

9.078.868.68

8.538.408.288.188.10

8.027.947.887.827.77

7.727.687.647.607.56

7.317.086.856.64

499999.0130.8118.0013.27

10.929.558.658.027.56

7.206.93

6.706.516.36

6.236.116.015.935.85

5.785.725.665.615.57

5.535.495.455.425.39

5.184.984.794.60

540399.1729.4616.6912.06

9.788.45.7.596.996.55

6.225.95

5.745.565.42

5.295.185.095.014.94

4.874.824.764.724.68

4.644.604.574.544.51

4.314.133.953.78

562599.2528.7115.9811.39

9.157.857.016.425.99

5.675.41

5.205.034.89

4.774.674.584.504.43

4.374.314.264.224.18

4.744.114.074.044.02

3.833.653.483.32

576499.3028.2415.5210.97

8.757.466.636.065.64

5.325.06

4.864.694.56

4.444.344.254.174.10

4.043.993.943.903.36

3.823.783.753.733.70

3.513.343.173.02

585999.9327.9115.2110.67

8.477.196.375.805.39

5.074.82

4.624.464.32

4.204.104.013.943.87

3.813.763.713.673.63

3.593.563.533.503.47

3.293.122.962.80

598199.3627.4914.8010.27

8.106.846.035.475.06

4.744.50

4.304.144.00

3.893.793.713.633.56

3.513.453.413.363.32

3.293.263.233.203.17

2.992.822.662.51

610699.4227.0514.379.89

7.726.475.675.114.71

4.404.16

3.963.803.67

3.553.453.373.303.23

3.173.123.073.032.99

2.962.932.902.872.84

2.662.502.342.18

623499.4626.6013.939.47

7.316.075.284.734.33

4.023.78

3.593.433.29

3.183.083.002.922.86

2.802.752.702.662.62

2.582.552.522.492.47

2.292.121.951.79

636699.5026.1213.469.02

6.885.654.864.313.91

3.603.36

3.163.002.87

2.752.652.572.492.42

2.362.312.262.212.17

2.132.102.062.032.01

1.801.601.381.00