Chapter 8:

Chapter 8:Analysis of Variance

Analysis of Variance-Analysis of Variance or ANOVA is a comparison test used to determine the significant difference among normal population means. The comparison in means of three (3) or more populations, which follow normal distributions, can be taken simultaneously in just one application of this test. This test is therefore a generalization of the z and t tests of two normal population means. This test was developed by Sir Ronald A. Fisher (1890-1962).

The following assumptions should be met in the use of ANOVA:The various groups are assumed to be with normal populations.The variance of the different groups is assumed to be equal.The random samples in the groups should be independent.

Formulas:Total sum of Squares (TSS)TSS=X - (X)/NWhere TSS= Total sum of squares X= Value of each entry N= Total number of items or entrySum of Squares Between-Columns (SSb)SSb = 1/No. of Rows (sum of each column) - (X)/NSum of Squares Within-Column (SS w) SS w = TSS- SSb

Mean Sum of Squares Between (MSSb) MSSb = SSb / dfb Where dfb = no. of columns 1Mean Sum of Squares Within (MSSw) MSSw = SSw/dfw Where dfw = (row*column) c 6. F-Value = MSSb / MSSw

Problem 10:Let us consider three groups of seven students, where each group is subjected to one of the three strategies or methods of teaching. Group A was exposed to Explanatory Approach, Group B for Cooperative Learning, and Group C for Traditional Method. The grades of the students are presented below. Test if there is a difference in the three methods or strategies of teaching at 5%level of significance.

StudentGroup AExplana-toryGroup BCoopera-tiveGroup CTradi-tionalGroup AGroup BGroup CXaXbXc(Xa)(Xb)(Xc)185861007225739610000290888981007744792139289888464792177444889087774481007569591878382817569688969388858649774472257899180792182816400

Steps:Ho: There is no significant difference among the three methods or strategies of teaching.= 5%Test statistic to be used: ANOVASolution: Compute for1. TSS = X - (X)/N = 164,887- (1859)/21= 164,887-164,565.76= 321.24

Where: X = (Xa)+(Xb)+(Xc) = 56384+54755+53748 = 164887 X = (Xa)+(Xb)+(Xc) = 628+619+612 = 18592. Sum of Squares Between-Column (SSb) SSb = 1/ No. of Rows (sum of each column) - (x)/N =1/7 (628+619+612) (1859)/21 = 164584.14-164565.76 = 18.38

3. Sum of Squares Within-Column(SSw)SSw = TSS-SSb = 321.24-18.38 = 302.86 4. Mean Sum of Squares Between(MSSb) MSSb = SSb / dfb = 18.38 / 2 = 9.19 Where: dfb = no. of columns 1 = 3-1 = 2

5. Mean Sum of Squares Within (MSSw) MSSw = SSw/dfw = 302.86/18 = 16. 825 Where: dfw = (row*column)-3 = (7x3)-3= 186. F-Value = MSSb / MSSw =9.19/16.83=0.546DF = 2 and 18 T.V. = 3.55

After the sum of squares have been computed, a summary table has to be presented:Accept Ho since the computed value of 0.546 is less than the tabular value of 3.55 Therefore, the three strategies of teaching are not significantly different from each other at an alpha of 5%.In as much as the result of the study is not significant, the researcher may stop at this point for his generalizations. But if the results of the study showed a significant result, the data will still be subjected for further testing to determine which of the pairs will show a significant difference in means.

Source of Variation Sum of SquaresDegrees of FreedomMean Sum of SquaresComputed FBetween ColumnWithin Column Total18.38302.86321.243-1=2(3x7)-3=18209.1916.830.546

Computations of ANOVA using MicroStatOne-way ANOVAGROUPMEANN 189.7147 288.4297 387.4297GRAND MEAN 88.52421SourceSum of SquaresD.F.Mean SquareF. RatioProb.BETWEEN18.38129.1900.5460.5884 WITHIN302.8571816.825TOTAL321.23820

From MegaStatOne factor ANOVAMeannStd. Dev.88.5238095289.772.69Group 188.5238095288.471.72Group 288.5238095287.476.35Group 388.5214.01TotalANOVA table SourceSSdfMSFp-valueTreatment18.3829.1900.55.5884Error302.861816.825Total321.2420

Exercise 22ANOVAName:Date:Course & Year:Score:Solve completely the following problems:Three brands of reducing pills were tried on a sample of 10 female adults; the data are reflected on the table below in terms of weight loss (lb) after a month of using these pills.

RespondentsBrand ABrand BBrand C123456788104.13.13.64.23.84.74.12.83.04.23.13.33.54.94.13.94.03.94.14.03.63.83.03.13.23.33.94.62.94.2

16

continuation.Test if there is a significant difference in the average weight loss (in lb) among the three groups of respondents using the three brands of reducing pills at 0.05 level of significance.

Exercise 22ANOVAName: Date:Course & Year:Score:2. Based from the survey results shown below, determine if there is a significant difference existing in the mean achievement of students from the three non-sectarian schools in Tuguegarao City by using ) 0.01 level of significance.Student No.School ASchool BSchool C12345678910768688908175878992858382858196798393899082908386928875897790

Exercise 22ANOVAName:Date:Course & Year:Score:3. The following are the heights in inches of six male college students of Cagayan Colleges Tuguegarao from the three regions of the country. Is there an evidence of height variation among these groups using the 0.05 level?Region 1Region 2Region 3123456635869726063576363696166757260596150

MEASURES OF CORRELATION-Correlation is a statistical tool to measure the association of two or more quantitative variables. It is concerned with the relationship in the change and movements of two variables. It is also defined as the measure of the linear relationship between two random variables x and y and is denoted by r. It measures the extent to which the points cluster about a straight line.

Three degrees of relationship or correlation between two variables Perfect correlation (positive and negative)Some degrees of correlation (positive and negative)No correlation

The quantitative interpretation of the degree of linear relationship existing is shown below. 1.00 Perfect Correlation (negative) correlation0.91 - 0.99 Very high positive (negative) correlation0.71 - 0.90 High positive (negative) correlation0.50 - 0.70 Moderately positive (negative) correlation, substantial relationship0.31 - 0.50 Low positive correlation (negative)0.01- 0.30 Slight correlation, negligible positive (negative) correlation0 No correlation

The five figures in the next page illustrate the degree of correlation between two variables.Figure A is a perfect positive correlation, which relates two variables whose values are both increasing.Figure B is a perfect negative correlation describes a situation where one variable increases, the other variable decreases.Figure C and D, some degree of positive or negative correlation which relates two variables whose value ranges from 0.1 - 0.99Figure E, No correlation, which describes a situation whose correlation coefficient is 0.

Summary of the different types or degrees of correlation between two variables

Figure A Figure BPerfect Positive Correlation Perfect Negative Correlation

Figure CFigure DFigure ESome Degree of Positive Correlation Some Degree of Negative CorrelationNo Correlation

23

Spearman Rank-It is used to determine the degree of relationship of two variables expressed as ORDINAL DATA.

Formula:

Rs = 1 - 6D2/ N3 - n

Problem 11: Randomly, selected jobs are ranked and stress. Does a significant relationship exist between salary and stress using 0.05 level of significance?

25

JobSalary RankStress RankDD2Lawyer2200Zoologist6711Doctor3639College Dean5411Hotel Manager7524Bank Officer10824Safety Inspec9900Police Officer81024Teacher4311Pilot1100D2=24

Solution: Rs = 1 6 D2 / N3 n= 1 6(24) / (10)3 10 = 0.145

From MegaStat:

Salary RankStress RankSalary Rank1.000Stress Rank.8551.00010Sample size .632Critical value .05(two-tail). 765Critical value .01(two-tail)

The Pearson Product-Moment Correlation Coefficient

The Pearson r from Raw ScoresFormula:

r = N XY - X Y / X2- (X) 2] [NY2 (Y)2] Where:

N= No. of Class

XY= Sum of the products of X and Y

X= Sum of X

Y= Sum of Y

X2= Sum of the squares of X

Y2= Sum of the squares of Y

Problem 12: Height and weight of 10 basketball players; find the Pearson r.

X (height in inches)Y(Weight in kilos)XYX2Y2

6565422542254225646440964096409678705460608449007271511251845041696544854761422566664356435643567068476049004624716948995041476170704900490049006771475744895041X=692Y=679XY=47,050X2=48,036Y2=46,169

Solution:

r= XY - X Y / X2- (X) 2] [NY2 (Y)2]

= 10(47050)-(692)(679) /2]-[10(46169)-(679) 2]

= 470500-469868 /

31

Continuation

= 632 /

= 632 /

= 632 / 985.35

=0.64

Interpretation: r= 0.64 is moderately positive correlation. There is substantially degree of correlation between the height and weight of 10 basketball players.

TESTING THE SIGNIFICANCE OF r-The test for significance of r is needed in order to know, whether the computed r is significant or not.Solutions:Ho: There is no significant relationship between height and weight of the 10 basketball players. Ha: There is a significant relationship between the height and weight of the 10 basketball players.2. Level of significance= 5%df=10-2=8Tabular Value= 1.859548

3. Test statistic to be used is t for r4. Compute for t:t = r / 1 r2 / n-2t= 0.64 / 1 (0.64)2 / 10 2t= 0.64 / 0.0738t= 0.64 / 0.271661554t= 2.36

5. Decision: The computed value 2.36 is greater the tabular value 1.8535. Hence, the null hypothesis is rejected.

6. Interpretation: There is a significant relationship between the height and weight of the 10 basketball players

REGRESION ANALYSISBivariate Linear Regression-Simple and multiple predictions are made with a technique called Regression Analysis.

Linear Regression Analysis-We now go beyond the notion of association and relation to try to examine (possible) casualty (or prediction). Sometimes, given information about one characteristics of a phenomenon, we can have some idea about the nature of another characteristics.

Continuation.. A statistical technique designed to predict values dependent variable from knowledge of the values of the one or more independent variable. It uses the principle of ordinary least squares where line is drawn through a scatter plot that minimizes the sum of squared residuals. In other words, a line is drawn as close as possible to all the cases in the sample. When one takes the values of X to estimate or predict corresponding Y values, the process is called simple prediction.

Continuation..Examples:We associate high caloric intake with body weight.If we know the temperature in Celsius, we can calculate the value in Fahrenheit.In Social Sciences, we infer the high income or high education lowers the desired family size.We can make these inferences, but we are not accurate. Therefore, regression is designed to help us to determine the probability that our inferences are sound. Put differently, it helps us to test the degree to which the dependent variable is affected by the independent variable.

GUIDELINES FOR USING LINEAR REGRESSION:If there is no significant linear correlation, do not use the regression equation to make predictions. When using the regression equation for predictions, stay within the scope of the available sample data. A regression equation based on old data is not necessarily valid now. Dont make predictions about a population that is different from the population from which the sample data were drawn.

The Basic Bivariate Regression EquationNon-Stochastic Equation-It is an error-free equation used to predict the value of y. It is an equation for perfect correlations.Formula: Y = a + bx (exact relationship)

Stochastic Equation-It is an equation where the estimate yields an error.-It is usually common in problems on social sciences.Formula: Y = a + bx (inexact relationship)

Where:

Y = independent or response variableX = independent or predictor variable (called explanatory or regressor variable)a = y-interceptb = slop of a linee = residual or error terme = Y

Where: = the estimated value of Y using the rergression equation

Formula:

a = ( Y) ( X2) (X) (XY) / N (X2) (X)2

b = N (XY) ( X) (Y) / N (X2) (X) 2

a = (679) (48036) (692) (47050) / 10 (48036) (692)2 = 38.666

b = 10 (47050) (692) (679) / 10 (48036) (692)2 = 0.4225

Scatter Plot-They provide a mean for visual inspection of data that a list of values for two variables cannot. They are essential for understanding the relationship between variables.

This scatter diagram makes 3 things clear:There seems to be a moderate positive relationship between x and y.

No straight line could be possibly drawn that would pass through all the point; most of the points would be above or below the best fitting line (least square regression line.)

If the relationship were a deterministic (r=1), the point would lie on the straight line.

Interpretation: Since probability value is less than alpha, the Ho is rejected. Therefore, there is a significant relationship between height and weight of the 10 basketball players.

Regression Analysisr20.411n 10r 0.641k 1Std. Error2.185Dep. VarY(weight in kilos)ANOVA tableSourceSSdfMSFp-valueRegression26.6995126.69955.59.0456Residual38.200584.7751Total64.90009Regression OutputConfidence IntervalVariablesCoefficientsStd. errort= (df=8)p-value95% lower95% upperIntercept38.665812.38253.123.014210.111767.2198X(height in inches)0.42250.17872.365.04560.01050.8344

Exercise 23Measures of CorrelationNameDateCourse and YearScoreSolve for the coefficient of correlation using the Pearson r formula or Spearman. Rank the following:Ten students were given tests in Statistics and English. The results are shown below:

StatisticsEnglish8790676067766189675890915078788992908788

Exercise 23Measures of CorrelationNameDateCourse and YearScore2. The table below shows how the nutrition experts and heads of household ranked 10 breakfast foods based on their palatability.

Nutrition ExpertsHeads of household3746718492101015235869

Exercise 23Measures of CorrelationNameDateCourse and YearScore

3. The 10 weeks sales of ABC Department Store in Tuguegarao City and its branch in Santiago City

Sales of ABC Store in Tuguegarao CitySales of ABC Store in Santiago City3171426073118243912223351950283555186339

Group 8Santiago, Jarys Christian C.Santos, AkieSarmiento, Lalli AnnaSeduguchi, KasumiValle, Coleen H.Vallente, AbiatharVillasper, ArbinSumayod, CressaLozada, Elijah

TABLE D. CRITICAL VALUES OF F

TABLE F. SPEARMAN RANK CORRELATION COEFFICIENT

TABLE E. PEARSON

Table B. Students t-Distributiondf0.400.250.100.050.0250.010.00510.3249201.0000003.0776846.31375212.7062031.8205263.6567420.2886750.8164971.8856182.9199864.302656.964569.9248430.2766710.7648921.6377442.3533633.182454.540705.8409140.2707220.7406971.5332062.1318472.776453.746954.6040950.2671810.7266871.4758842.0150482.570583.364934.0321460.2648350.7175581.4397561.9431802.446913.142673.7074370.2631670.7111421.4149241.8945792.364622.997953.4994880.2619210.7063871.3968151.8595482.306002.896463.3553990.2609550.7027221.3830291.8331131.262162.821443.24984100.2601850.6998121.3721841.8124612.228142.763773.16927110.2595560.6974451.3634301.7958852.200992.718083.10581120.25903306954831.3562171.7822882.178812.681003.05454130.2585910.6938291.3501711.7709332.160372.650313.01228140.2582130.6924171.3450301.7709332.144792.624492.97684150.2578850.6911971.3450301.7530502.144792.602482.94671

55

df0.40.250.100.050.0250.010.005160.2578850.6911971.3406061.7530502.131452.602482.94671170.2573470.6891951.3333791.7396072.109822.566932.89823180.2571230.6883641.3303911.7340642.100922.552382.87884190.2569230.6876211.3277281.7291332.093022.539482.86093200.2567430.6869541.3253411.7247182.085962.527982.84534210.2565800.6863521.3231881.7207432.079612.517652.83136220.2564320.6858051.3212371.7171442.073872.508322.81876230.256970.6853061.3194601.2138722.068662.499872.80734240.2561730.6848501.3178361.7108822.063902.492162.79694250.2560600.6844301.3163451.7081412.059542.485112.78744260.2559550.6340431.3149721.7056182.055532.478632.77871270.2558580.6836851.3137031.7032882.051832.472662.77068280.2557680.6833531.3125271.7011312.048412.467142.76326290.2556840.6830441.3114341.6991272.045232.467142.75639Large0.02533470.6744901.2815521.6448541.959962.326352.57583

Df0.950.900.700.500.200.100.050.020.0110.003930.01580.1480.4551.6422.7063.8415.4126.63520.1030.2110.7131.3863.2194.6055.9917.8249.21030.3520.5841.4242.3664.6426.2517.8159.83711.34640.7111.0642.1953.3575.9897.7799.48811.66813.27751.1451.6103.0004.3517.2899.23611.07013.38815.08661.6352.2043.8285.3488.55810.64512.59215.03316.81272.1672.8334.6716.3469.80312.01714.06716.62218.47582.7333.4905.5277.34411.03013.36215.50718.16820.09093.3254.1686.3938.34312.24214.68416.91919.67921.666103.9404.8657.2679.34213.44215.98718.30721.16123.209

114.5755.5788.14610.34114.63117.27519.67522.61824.725125.2266.3049.03411.3015.81218.54921.02624.05426.217135.8927.0429.92612.34016.98519.81222.36225.47227.688146.5717.79010.82113.33918.15121.06423.68526.87329.141157.2618.544711.72114.33919.3112.30724.99628.25930.578167.9629.31212.62415.33820.46523.54226.29629.63332.000178.67210.08513.53116.33821.61524.76927.58730.99533.409189.39010.86514.44017.33822.76025.98928.86932.34634.8051910.11711.65115.35218.33823.90027.20430.14433.68736.1912010.85112.44316.26619.33725.03828.41231.41035.02027.566

2111.59113.24017.18220.33726.17129.61532.67136.34338.9322212.33814.04118.10121.33727.30130.81333.92437.65940.2892313.09114.84819.02122.33728.42932.00735.17238.96841.6382413.84815.65919.94323.33729.55333.19636.41540.27042.9802514.61116.47320.56724.33730.67534.38237.65241.56644.3142615.37917.29221.79225.33631.79535.56338.88542.85645.6422716.15118.11422.71926.33632.91236.74140.11344.14046.9632816.92818.93923.64927.33634.02737.91641.33745.41948.2782917.70719.75824.37728.33635.13939.08742.55746.69349.5883018.40320.59925.50829.33636.25040.25643.77347.96250.892

Sheet1Degrees of Freedom for Greater Mean Square1234567121 161.45 4052.10199.50 4999.03215.72 5403.49224.57 5625.14230.17 5764.08233.97 5859.39238.89 5981.34243.92 6105.83254.32 6366.48218.51 98.4919.00 99.0119.16 99.1719.25 99.2519.30 99.3019.33 99.3319.37 99.3619.41 99.4219.50 99.50310.13 34.129.55 30.819.28 29.469.12 28.719.01 28.248.94 27.918.84 27.498.74 27.058.53 26.1247.71 21.206.94 18.006.59 16.696.39 15.986.26 15.526.16 15.216.04 14.805.91 14.375.63 13.4656.61 16.265.79 13.27 5.41 12.065.19 11.395.05 10.974.95 10.674.82 10.274.68 9.894.36 9.0265.99 13.74 5.14 10.924.76 9.784.53 9.154.39 8.754.28 8.474.15 8.104.00 7.723.67 6.8875.59 12.254.74 9.554.35 8.454.12 7.853.97 7.463.87 7.193.73 6.843.57 6.473.23 5.6585.32 11.264.46 8.654.07 7.593.84 7.013.69 6.633.58 6.373.44 6.033.28 5.672.93 4.8695.12 10.564.26 8.023.86 6.993.63 6.423.48 6.063.37 5.803.23 5.473.07 5.112.71 4.31104.96 10.044.10 7.563.71 6.553.48 5.99 3.33 5.643.22 5.393.07 5.062.91 4.712.54 3.91

Sheet1Degrees of Freedom for Greater Mean Square123456712104.96 10.044.10 7.563.71 6.553.48 5.993.33 5.643.22 5.393.07 5.062.91 4.712.54 3.91114.84 9.653.98 7.203.59 6.223.36 5.673.20 5.323.09 5.072.95 4.742.79 4.402.40 3.60124.75 9.333.88 6.93 3.49 5.953.26 5.413.11 5.063.00 4.822.85 4.502.69 4.162.30 3.36134.67 9.073.80 6.703.41 5.743.18 5.203.02 4.862.92 4.622.77 4.302.60 3.9622.10 3.16144.60 8.863.74 6.513.34 5.563.11 5.032.96 4.692.85 4.462.70 4.142.53 3.802.13 3.00154.54 8.683.68 6.363.29 5.423.06 4.892.90 4.562.79 4.322.64 4.002.48 3.672.07 2.87164.49 8.533.63 6.233.24 5.293.01 4.772.85 4.442.74 4.202.59 3.892.42 3.552.01 2.75

Sheet1Degrees of Freedom for Greater Mean Square123456712174.45 8.403.59 6.113.20 5.182.96 4.672.81 4.342.70 4.102.55 3.792.38 3.451.96 2.65184.41 8.283.55 6.013.16 5.092.93 4.582.77 4.252.66 4.012.51 3.712.34 3.371.92 2.57194.38 8.183.52 5.933.13 5.012.90 4.502.74 4.172.63 3.942.48 3.632.31 3.301.88 2.49204.35 8.103.49 5.553.10 4.942.87 4.432.71 4.102.60 3.592.45 3.562.28 3.231.84 2.42214.32 8.103.47 5.783.07 4.872.84 4.372.68 4.042.57 3.812.42 3.512.25 3.171.81 2.36224.30 7.943.44 5.723.05 4.822.82 4.322.23 3.122.66 3.992.55 3.752.40 3.451.78 2.30

Sheet1N0.050.014150.9160.8290.94370.7140.89380.6430.833

90.60.783100.5640.746120.5060.712140.4560.645160.4250.601

180.3990.564200.3770.534220.3590.508240.3430.485260.3290.465

280.3170.448300.3060.432

Sheet1DF0.050.0110.996920.999987720.950.9930.87830.9587340.81140.917250.75450.8745

60.70670.834370.66640.797780.63190.764690.60210.7348100.5760.7079

110.55290.6835120.53240.6614130.51390.6411140.49730.6226150.48210.6055

160.46830.5897170.45550.5741180.44380.5614190.43290.5487200.42270.5368

Sheet1DF0.050.01210.38090.4869220.34940.4487230.32460.418240.30440.3932250.28750.3721

260.27320.3541270.250.3245280.23190.3017290.21720.283300.205.26.73

Sheet1

23 4.28 3.42 3.03 2.80 2.64 2.53 2.38 2.20 1.76 7.88 5.66 4.74 4.26 3.94 3.71 3.41 3.07 2.26 23 4.28 3.42 3.03 2.80 2.64 2.53 2.38 2.20 1.76 7.88 5.66 4.74 4.26 3.94 3.71 3.41 3.07 2.26 24 4.26 3.40 3.01 2.78 2.62 2.51 2.36 2.18 1.73 7.82 5.61 4.72 4.22 3.90 3.67 3.36 3.03 2.21 25 4.24 3.38 2.99 2.76 2.60 2.49 2.34 2.16 1.71 7.77 5.57 4.68 4.18 3.86 3.63 3.32 2.99 2.17 26 4.22 3.37 2.98 2.74 2.59 2.47 2.32 2.15 1.69 7.72 5.53 4.64 4.14 3.82 3.59 3.29 2.96 2.13 27 4.21 3.35 2.96 2.73 2.57 2.46 2.30 2.13 1.67 7.68 5.49 4.60 4.11 3.78 3.56 3.26 2.93 2.10 28 4.20 3.34 2.95 2.71 2.56 2.44 2.29 2.12 1.65 7.64 5.45 4.57 4.07 3.75 3.53 3.23 2.90 2.06 29 4.18 3.33 2.93 2.70 2.54 2.43 2.28 2.10 1.64 7.60 5.42 4.54 4.04 3.73 3.50 3.20 2.28 2.03 30 4.17 3.32 2.92 2.69 2.53 4.24 2.27 2.09 1.62 7.56 5.39 4.51 4.02 3.70 3.47 3.17 2.84 2.01

Sheet1 35 4.12 3.26 2.87 2.64 2.48 2.37 2.22 2.04 1.57 7.42 5.27 4.40 3.91 3.59 3.37 3.07 2.74 1.90 40 4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.00 1.52 7.31 5.18 4.31 3.83 3.51 3.29 2.99 2.66 1.82 45 4.06 3.21 2.81 2.58 2.42 2.31 2.15 1.97 1.48 7.23 5.11 4.25 3.77 3.45 3.23 2.94 2.61 1.75 50 4.03 3.18 2.79 2.56 2.40 2.29 2.13 1.95 1.44 7.17 5.06 4.20 3.72 3.41 3.19 2.89 2.56 1.68 60 4.00 3.15 2.76 2.52 2.37 2.25 2.10 1.92 1.39 7.08 4.98 4.13 3.65 3.34 3.12 2.82 2.52 1.60 70 3.98 3.13 2.74 2.50 2.35 2.23 2.07 1.89 1.35 7.01 4.92 4.07 3.60 3.29 3.07 2.78 2.45 1.53 80 3.96 3.11 2.72 2.49 2.33 2.21 2.06 1.88 1.31 6.96 4.88 4.04 3.56 3.26 3.04 2.74 2.42 1.47

Sheet1 90 3.95 3.10 2.71 2.47 2.32 2.20 2.04 1.86 1.28 6.92 4.85 4.01 3.53 3.23 3.01 2.72 2.39 1.43 100 3.94 309 2.70 2.46 2.30 2.19 2.03 1.85 1.26 6.90 4.82 3.98 3.51 3.21 2.99 2.69 2.37 1.39 125 3.92 3.07 2.68 2.44 2.29 2.17 2.01 1.83 1.21 6.84 4.78 3.94 3.47 3.17 2.95 2.66 2.33 1.32 200 3.89 3.04 2.65 2.42 2.26 2.14 1.98 1.80 1.14 6.76 4.71 3.88 3.41 3.11 2.89 2.60 2.28 1.21 300 3.87 3.03 2.64 2.41 2.25 2.13 1.97 1.79 1.10 6.72 4.68 3.85 3.38 3.08 2.86 2.57 2.24 1.14 400 3.86 3.02 2.63 2.40 2.24 2.12 1.96 1.78 1.07 6.70 4.66 3.83 3.37 3.06 2.85 2.56 2.23 1.11 500 3.86 3.01 2.62 2.39 2.23 2.11 1.96 1.77 1.06 6.69 4.65 3.82 3.36 3.05 2.84 2.55 2.22 1.08 1000 3.85 3.00 2.61 2.38 2.22 2.10 1.95 1.76 1.03 6.66 4.63 3.80 3.34 3.04 2.82 2.53 2.20 1.04 * 3.84 2.99 2.60 2.37 2.21 2.09 1.94 1.75 6.64 4.60 3.78 3.32 3.02 2.80 2.51 2.18