SMTX1014- PROBABILITY AND STATISTICS UNIT V ANALYSIS OF VARIANCE

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1 SMTX1014- PROBABILITY AND STATISTICS UNIT V ANALYSIS OF VARIANCE

2 STATISTICAL QUALITY CONTROL Introduction and Process Control Control Charts for X and R Control Charts for X and S p Chart np Chart c Chart

3 t- TEST F-TEST ONE WAY CLASSIFICATION TWO WAY CLASSIFICATION

4 F-test (Test of significance of the difference between population variances) Use of F-test: F test is used to the equality of variance of the population which two small samples have been drawn. Test statistic σ 2 1 σ 2 2 F estimated estimated var of var of sample1 sample2

5 2 1 n1s n 1 n and 2 1 n2 2 s2 1 Where n 1 sample size 1 n 2 - sample size 2 s 1 standard deviation of sample 1 s 2 standard deviation of sample 2 Degrees of freedom 1 n1 1; 2 n2 1 Note : we should always make F>1. i.e F greater variance smaller variance

6 CF = GT 2 / N Q = ΣΣ X ij2 CF Q 1 = Σ [CT 2 / R] CF Q 2 =Q-Q 1 One way Classification Source of Variance Sum of Squares Degrees of freedom Mean Squares F Ratio Between Samples(Columns) Q 1 SSC c-1 MSC=Q 1 /c-1 F=MSC/MSE (Greater/smaller) Within Samples (Rows) Q 2 SSE n-c MSE=Q 2 /n-c

7 Where c-no of columns n-given no of observations MSC Mean squares columns MSE - Mean squares error CF correction factor GT Grand total CT - column total SSC-Sum of squares columns SSE-Sum of squares Rows Problems on one way classification 1)Setup ANOVA for the following per hectare yield for three varieties of Wheat, Each grown in five plots. Test whether is significant difference among the average yields in the 3 varieties of wheat. Test the hypothesis that the population are equal at 5% level of significance

8 Variety of Wheat plot A 1 A 2 A Solution: H 0 : There is no significance difference between the samples X 1 X 2 1 X 2 X 2 2 X 3 X

9 CF = GT 2 / N =(120) 2 /15 = 960 Q = ΣΣ X ij2 CF = ( )-960 =100 Q 1 = Σ [CT 2 / R] CF =[40 2 / / /5]-960=( 5000/5) -960 = 40 Q 2 =Q-Q1 = = 60 Source of Variance Sum of Squares Degrees of freedom Mean Squares F Ratio Between Samples(Columns) Q 1 =40 SSC c-1=3-1=2 MSC=Q 1 /c-1 =40/2= 20 F=MSC/MSE (Greater/smaller) =20/4 =5 Within Samples (Rows) Q 2 =60 SSE n-c =15-3=12 MSE=Q 2 /n-c 50/12=5 Calculated value F = 5 > Table value 2,12 at 5% level is 3.88 H 0 is rejected. Result: There is a difference between the samples

10 Problems for practice on one way classification 1.Three processes A,B, and C are tested to see whether their outputs are equivalent. The following observation of output are made. A B C

11 2.For salesmen who served four different areas sold the units as follows. Is there any significant difference in their performance. A B C D The following table shows the lives in hours of four brands of electric lamps. Brands A Brands B Brands C Brands D Perform an analysis of variance test homogeneity of the mean lives of four brands of lamps

12 CF = GT 2 / N Q = ΣΣ X ij2 CF Q 1 = Σ [CT 2 / R] CF Q 2 = Σ [RT 2 / C] CF Q 3 =Q-Q 1 -Q 2 Two way Classification Source of Variance Between Samples (columns) Between Samples Sum of Squares Q 1 SSC Q 2 SSR Rows Q Error 3 SSE Degrees of freedom Mean Squares F Ratio C-1 MSC=Q 1 /K-1 F=MSC/MSE (Greater/smaller) F=MSR/MSE r-1 MSR=Q 2 /n-k (C-1)(r-1) MSE=Q 3 /(c-1)(r-1)

13 Problems on Two way classification 1)A tea company appoints four Salesmen A,B,C and D and observes their sales in three seasons-summer, winter and monsoon. The out sales in 1000 of units given below Seasons A B C D Summer Winter Monsoon

14 Solution: X 1 X 2 X 3 X 4 RT RT 2 /4 X 1 2 X 2 2 X 3 2 X /4 = / / / /

15 CF = GT 2 / N = (492) 2 /12 = Q = ΣΣ X ij2 CF =( ) =132 Q 1 = Σ [CT 2 / R] CF = [( = )-20172]=( )=42 Q 2 = Σ [RT 2 / C] CF =[( )-20172=26 Q 3 =Q-Q 1 -Q 2 = 64 MSC =42/3 = 14 MSR =26/2 = 13 MSE(Residual) = 64/6 = F=1.313 F=1.219 Table values at 3,6 is 9.78 > ,6 is 10.92>1.219

16 Problems for practice on Two way classification 1.An Experiment was designed to study the performance of 4 different detergents for cleaning fue injectors. The following cleanness readings were obtained with specially designed equipment for 12 tanks of gas distributed over 3 different model of engines Engine 1 Engine 2 Engine 3 DetergentA DetergentB DetergentC DetergentD

17 Looking on the detergents of treatments and the Engines at blocks, Obtain the appropriate ANOVA table and test at 1% level of significance 2. Perform the two way classification for the following Plots of land Treatment A B C D Seasons total Land1 Land2 Land

18 INTRODUCTION Quality Statistical Quality Control Quality Control Charts Process Control Product Control Variables Attributes

19 INTRODUCTION In these days of tough business competition, it has become essential to maintain the quality of the goods manufactured and market them at reasonable price. If the consumers feel satisfied with regard to the quality, price, etc. of the product manufactured by a certain company, it will result in goodwill for the product and in increase in sales.

20 If not and if proper attention is not given to the complaints of the consumers regarding quality, the manufacturer cannot push through his products in the market and ultimately he has to quit the market. Hence it is important to maintain and improve the quality of the manufactured products for the manufacturer to remain and flourish in his business.

21 Quality Goods are said to be of good quality if they satisfy the consumer. Goods are said to be of good quality if they meet the expected functional use. The quality not only measures the level of satisfaction in meeting the customers needs but also is a function of time period in which it continues to meet customers needs.

22 INTRODUCTION (CONTINUED ) Statistical Quality Control (SQC) SQC is a statistical method for finding whether the variation in the quality of the product is due to random causes or assignable causes. SQC does not involve inspecting each and every item produced for quality standards, but involves inspection of samples of items produced and application of tests of significance.

23 ARIOUS CHARTS OF QUALITY CONTROL Process Control Control Charts Variables Attributes X - Chart S - Chart R - Chart p - Chart np- Chart c - Chart

24 Process Control The control of the quality of the goods while they are in the process of production. In process control, we could detect the defects and faults of the process and could correct the errors. Control Chart Control chart is a graphical device mainly used for the study and control of the manufacturing process. Control chart is also called shewhart chart.

25 Types of Control Charts Control charts of variables Control charts of attributes Variables The quality characteristics of a product that are measurable. Example Diameter of wires Life of an electric bulb Setting time of concrete

26 Attributes The quality characteristics of a product that are not measurable. Example Visual defects found during inspection Coloured threads in white cloth Bubbles of air in windscreen Clerical errors in an invoice

27 CONTROL CHART Out of control Upper Control Limit (UCL) Process Average (CL) Lower Control Limit (LCL) CONTROL CHART Sample number

28 Advantages of Control Chart Control chart, helps us to rectify the faults and errors during the process or even after the process is over. By quality control methods, contain and hold the variability in the production process due to chance variation only. Any change of process in the production line can be tested very easily.

29 Advantages of Control Chart (Continued ) Warning can be taken when the sample points lie outside warning limits and between action or control limits. Since the testing is done by sampling methods based on probability, there is a lot of saving in time and cost. Decisions can be taken with more reliability and confidence. Also by use of control charts, wastage, expenditure and spoilage can be minimized.

30 Control limits for X 1 n X i 1 R R i n CL X X chart LCL X A2 R UCL X A2 R Control limits for CL R R LCL D3 R UCL D4R chart

31 c Chart c where 1 N c i CL y c LCL c 3 c UCL c 3 c p c constant size. Chart is average number of defects or flaws in sample of 1 np np n CL y p LCL p 3 p(1 p) n

32 (1 ) 3 p UCL p p n n where is average sample size, is average value of p. np Chart p 1 np np N CL y np LCL np 3 np(1 p) UCL np 3 np(1 p) where n is sample size, p is proportion defective p is average value of p.

33 X and R Chart Problems: A machine fills boxes with dry cereal. 15 samples of 4 boxes are drawn randomly. The weight of the sampled boxes are shown as follows. Draw the control charts for the sample mean and sample range and determine whether the process is in a state of control.

34 Sample Weight of Boxes Number

35 Solution: Sample Weight of Boxes Number Mean Range Average

36 X 1 1 X i ( ) n R R i ( ) 1.33 n 15 From the table of controlchart constants,for thesamplesizen = 4, A D3 0 D Control Limits for X-Chart CL X LCL X A2 R (1.33) UCL X A R (1.33)

37 Control Limits for R -Chart CL R 1.33 LCL D3R 0 UCL D4R 2.282(1.33) 3.04 Conclusion Since all the sample points lie within upper and lower line both X bar and R chart, the process in under control.

Sample Mean 13 X-Bar Chart 12.5 UCL, 12.53 12 11.5 CL, 11.

38 Sample Mean 13 X-Bar Chart 12.5 UCL, CL, LCL,

39 Sample Range 3.5 R- Chart 3 UCL, CL, LCL,

40 EXERCISE 1. A food company puts mango juice in cans, each of which is advertised to contain 10 ounces of the juice. The weights of the juice drained from cans immediately after filling 20 samples each of 4 cans are taken by random sampling method ( at an interval of 30 minutes) and given in the following table in units of 0.01 ounce in excess of 10 ounces. To control the excess weights of mango juice drained while filling, draw the X bar chart and R chart and comment on the nature of control.

Sample Number Weights Drained 1 2 3 4 1 15 12 13 20 2 10 8 8 14 3 8 15 17 10 4 12 17 11 12 5 18 13 15 4 6 20 16 14 20 7 15 19 23 17 8 13 23 14 16 9 9 8 18

41 Sample Number Weights Drained

42 EXERCISE 2. The following data give the coded values of the crushing strengths of concrete blocks obtained from 20 samples each of size 5. Draw the X bar chart and R chart and comment on the state of control.

43 Sample Number Values of X

44 X and S Chart Problem The following data give the coded measurements of 10 samples each of size 5, drawn from the production process at intervals 1 hour. Calculate the sample mean and S.D s and draw the Sample control Number chart for X and S Chart. Coded Measurements(X)

45 Sample Number Solution Coded Measurements(X) Mean S.D We first compute the mean and S.D for each sample Average

46 Now X 1 1 X i ( ) 11 n s s i ( ) 2.32 n 10 From the table of control chart constants,for thesamplesize n=4, we have A B 3 0 B

47 ControlLimits for X -Chart CL X 11 n 5 LCL X A 1 s 11 (1.596) (2.59) 6.86 n 1 4 n 5 UCL X A 1 s 11 (1.596) (2.32) n 1 4 Control Limits for s - Chart CL s 2.32 LCL B 3 s 0 UCL B s 2.089(2.32)

48 Sample Mean X- Bar Chart UCL, Mean, CL, LCL,

49 Sample S.D(s) 6 s-chart 5 UCL, CL, LCL,

50 Conclusion The given mean values lies between 6.86 and and the given S.D values lies between 0 and Hence the process is under control with respect to average and variability.

51 EXERCISE 1. The values of sample mean X bar and sample standard deviation s for 15 samples, each of size 4, drawn from a production process are given below draw the appropriate control charts for the process average and process variability. Comment on the state of control.

52 Sample Number Mean S.D

53 EXERCISE 2. The following table gives the sample mean X bar and sample standard deviation s for 15 samples, each of size 4, in the production of a certain component. Draw X bar and s charts and comment on the state of control.

54 Sample Number Mean S.D

55 p Chart Problems 15 samples of 200 items each were drawn from the output of a process. The number of defective items in the samples are given below, prepare a control chart for the fraction defective and comment on the state of control Sample No No. of defective

56 Sample No. Solution: No. of defective (np) We first compute the fraction defectives Fraction defectives

57 Now np ) 180 np np 12 n 15 p 12 ( Q each samplecontains 200 items) For the p-chart CL LCL p 0.06 p(1 p) p (3) 0.01 n 200 UCL p(1 p) p (3) 0.11 n 200

58 Fraction defective 0.12 p-chart UCL, CL, LCL, Conclusion Since all the sample points lies between the LCL and UCL lines, the process is under control.

59 EXERCISE 1. Fifteen samples each of size 50 were inspected and the number of defectives in the inspection were 2, 3, 4, 2, 3, 0, 1, 2, 2, 3, 5, 5, 1, 2, 3. Draw the control chart for the number of defectives and comment on the state of control. 2. On inspection of 10 samples, each of size 400, the numbers of defective articles were 19, 4, 9, 12, 9, 15, 26, 14, 15, 17. Draw the n chart and comment on the state of control.

60 np Chart Problems In a factory producing spark plugs, the number of defectives found in the inspection of 15 lots of 100 each is given below. Draw the control chart for the number of defectives and comment on the state of control. Sample No No. of defective

61 Now np ) 90 np 1 90 np 6 N 15 6 p ( Q each samplecontains100items) For the np -Chart CL np 6 LCL np 3 np(1 p) 6 (3) LCL 0( cannot be nagative) UCL np 3 np(1 p) 6 ( 3)

62 No of defective np-chart Y=13.12(UCL) Y=6(CL) Y=0(LCL) Conclusion Since all the sample points lies between the LCL and UCL lines, the process is under control.

63 Fraction defective np-chart UCL, CL, LCL, Conclusion Since all the sample points lies between the LCL and UCL lines, the process is under control.

64 EXERCISE 1. Using the following data, construct the np - chart and comment on the state of control. Assume that 200 items are inspected each day. 2. Day On inspection 1 2 of samples, 5 6 each 7 8of size , 11the 12numbers of 15 No. defective of articles were 19, 4, 9, 12, 9, 15, 26, 14, 15, 17. Draw the np defective chart and comment on the state of control.

65 c Chart Problems A plant produces paper for newsprint and rolls of paper are inspected for defects. The results of inspection of 20 rolls of papers are given below. Draw the c chart and comment on the state of control. Roll No No. of defective Roll No No. of defective

66 Now c i 220 c c i 11 N 20 For the c - Chart CL c 11 LCL c 3 c 11 (3) UCL c 3 c 11 (3) Conclusion Since one point falls outside the control lines the process is out of control.

67 Number of Defects (c) c chart Y=20.95 (UCL) Y=11 (CL) Y=1.05 (LCL)

EXERCISE 1. 15 tape recorders were examined for quality control test. The number of defects in each tape recorder is recorded below.

68 EXERCISE tape recorders were examined for quality control test. The number of defects in each tape recorder is recorded below. Draw the appropriate control chart and comment on the state of control. Unit No No. of defective Unit No No. of defective

69 EXERCISE 2. Twenty half litre milk bottles are selected at random from a process and the numbers of air bubbles observed from the bottles are given in the following table. Draw the approximate control chart and comment on the nature of Bottle control. No No. of defective Bottle No No. of defective

Statistical quality control (SQC)

Statistical quality control (SQC) The application of statistical techniques to measure and evaluate the quality of a product, service, or process. Two basic categories: I. Statistical process control (SPC):