{"id":281,"date":"2018-10-31T04:59:42","date_gmt":"2018-10-31T04:59:42","guid":{"rendered":"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=281"},"modified":"2018-10-31T06:07:01","modified_gmt":"2018-10-31T06:07:01","slug":"analysis-of-variance-and-experimental-design-testing-for-equality-of-k-population-means","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/chapter\/analysis-of-variance-and-experimental-design-testing-for-equality-of-k-population-means\/","title":{"rendered":"Analysis of Variance and Experimental Design: testing for equality of k population means"},"content":{"raw":"<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Introduction<\/strong>\r\n<ul>\r\n \t<li>Experimental Design of Analysis of Variance<\/li>\r\n \t<li>Types of Experimental Design and Analysis and Variance<\/li>\r\n \t<li>Assumptions for ANOVA<\/li>\r\n \t<li>Steps of solving the experimental design<\/li>\r\n \t<li>Test for the Equality of <em>k<\/em> Population Means<\/li>\r\n \t<li>Test for the Equality of <em>k<\/em> Population Means: An Observational Study<\/li>\r\n \t<li>Summary<\/li>\r\n \t<li>Self-Check Exercise<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Quadrant-I<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Analysis of Variance and Experimental Design: testing for equality of k population means<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Learning Objectives:<\/strong>\r\n<ul>\r\n \t<li>After the completion of this module the student will understand:<\/li>\r\n \t<li>Experimental Design of Analysis of Variance<\/li>\r\n \t<li>Types of Experimental Design and Analysis and Variance Assumptions for ANOVA<\/li>\r\n \t<li>Steps of solving the experimental design<\/li>\r\n \t<li>Test for the Equality of <em>k<\/em> Population Means<\/li>\r\n \t<li>Test for the Equality of <em>k<\/em> Population Means: An Observational Study<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong>Introduction of ANOVA<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The analysis of variance frequently referred to by the contraction ANOVA is a statistical technique specially designed to test whether the means of more than TWO quantitative populations are equal.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">This technique developed by R.A. FISHER. In 1920 is capable of fruitful application to a diversity of practical problems. Basically it consists of classifying and cross-classifying statistical results and testing whether the means of a specified population differ significantly.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Introduction to Experimental Design and Analysis of Variance<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The statistical studies can be classified as being either experimental or observational. In an experimental study, one or more factors are controlled so that data can be obtained about how the factors influence the variables of interest but as far as in an observational study, no attempt is made to control the factors. One of the important points is that in this technique cause and effect relationship are easier to establish in experimental studies than in observational studies.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Analysis of variance (ANOVA) is used to analyze the data obtained from experimental or observational studies. In this study factor consider as a variable that the experimenter has selected for investigation also a treatment is a level of a factor it can be explained by an example, if location is a factor, and then a treatment of location can be Delhi, Ghaziabad, and Noida. So it concludes that experimental units are the objects of interest in the experiment. A completely randomized design is an experimental design in which the treatments are randomly assigned to the experimental units.<\/p>\r\n&nbsp;\r\n\r\n<strong>Types of Experimental Design and Analysis and Variance<\/strong>\r\n\r\n&nbsp;\r\n\r\nThree types of experimental designs are introduced.\r\n<ul>\r\n \t<li>A completely randomly design<\/li>\r\n \t<li>A randomized block design<\/li>\r\n \t<li>A factorial experiment<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong>Assumptions for ANOVA<\/strong>\r\n\r\n&nbsp;\r\n\r\nFor each population, the response (dependent) variable is <em>normally distributed<\/em>.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The variance of the response variable, denoted <em>s<\/em> 2, is the <em>same<\/em> for all of the populations. The observations must be <em>independent<\/em>.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Steps for calculation<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Between-Treatments Estimate of Population Variance <em>s<\/em> <\/strong><strong>2<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The estimate of <em>s<\/em> 2 based on the variation of the sample means is called the mean square due to treatments and is denoted by MSTR.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter wp-image-285\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-136.png\" alt=\"\" width=\"445\" height=\"158\" \/>\r\n\r\nk is the number of treatments (total of samples)\r\n\r\n<span style=\"font-size: 1em\">n <sub>j<\/sub> is the number of observations in treatment j<\/span>\r\n\r\nx<sub>j\u00a0\u00a0<\/sub>is\u00a0the sample mean of treatment j\r\n\r\n<del>x<\/del> is the overall mean, i.e. the average value of ALL the observations from all the treatments\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-286\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-137.png\" alt=\"\" width=\"153\" height=\"157\" \/>\r\n\r\n<strong>Within-Treatments Estimate of Population Variance <em>s<\/em> <\/strong><strong>2<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The estimate of <em>s<\/em> 2 based on the variation of the sample observations within each sample is called the mean square due to error and is denoted by MSE.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-287\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-138.png\" alt=\"\" width=\"221\" height=\"115\" \/>\r\n\r\nWhereas, MSE is mean square due to error\r\n\r\n&nbsp;\r\n\r\n<strong>Test for the Equality of <em>k<\/em> Population Means<\/strong>\r\n\r\n&nbsp;\r\n\r\nHypothesis\r\n\r\n&nbsp;\r\n\r\n<em>H<\/em>0 (null hypothesis):<em>\u00a0 m<\/em>1 =<em> m<\/em>2 =<em> m<\/em>3 = . . . =<em> m<\/em>k\r\n\r\n&nbsp;\r\n\r\n<em>H<\/em>a (alternative hypothesis): Not all population means are equal\r\n\r\n&nbsp;\r\n\r\nTest Statistic\r\n\r\n&nbsp;\r\n\r\n(MSTR= Mean square due to treatment, MSE= Mean square due to error)\r\n\r\n&nbsp;\r\n\r\nRejection Rule\r\n\r\n&nbsp;\r\n\r\n<em>p<\/em>-value Approach:\u00a0 Reject<em> H<\/em>0 if<em> p<\/em>-value &lt;<em> a<\/em>\r\n\r\n&nbsp;\r\n\r\nCritical Value Approach: Reject <em>H<\/em>0 if <em>F<\/em> &gt; <em>F<\/em><em>a<\/em>\r\n\r\n&nbsp;\r\n\r\nWhereas the value of <em>F<\/em><em>a<\/em> is based on an <em>F<\/em> distribution with <em>k<\/em> - 1 numerator d.f. and <em>n<\/em>T - <em>k<\/em> denominator d.f.\r\n\r\n&nbsp;\r\n\r\n<strong>6.\u00a0<\/strong><strong>Test for the Equality of <em>k<\/em> Population Means: An Observational Study<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n1.\u00a0 \u00a0Determine one estimate of the population variance from the <strong>'variance among<\/strong> <strong>the sample means'<\/strong>.\r\n\r\n&nbsp;\r\n\r\n2.\u00a0 \u00a0Determine a second estimate of the population variance from the <strong>'variance<\/strong> <strong>within the samples'<\/strong>.\r\n\r\n&nbsp;\r\n\r\n3.\u00a0\u00a0<strong>Compare these two estimates<\/strong>: If they are approximately equal in value, accept the null hypothesis.\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-288\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-139.png\" alt=\"\" width=\"374\" height=\"164\" \/>\r\n\r\n&nbsp;\r\n\r\nwhere <em>x<\/em>\u00a0 = sample mean\r\n\r\nx\u00a0\u00a0 = grand mean\r\n\r\nk = number of samples\r\n\r\n<img class=\"aligncenter size-full wp-image-290\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-141.png\" alt=\"\" width=\"311\" height=\"44\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNumber of degree of freedom in the numerator of the F ratio = (number of samples - 1).\r\n\r\nNumber of degree of freedom in the denomination of the F ratio = (nj\u20131) = nT-K\r\n\r\n<span style=\"font-size: 1em\">n<\/span><sub>j<\/sub><span style=\"font-size: 1em\"> = size of jth sample n<\/span><sub>T<\/sub><span style=\"font-size: 1em\"> =\u00a0 n<\/span><sub>j<\/sub><span style=\"font-size: 1em\"> = total sample size<\/span>\r\n\r\n<span style=\"font-size: 1em\">k = number of samples<\/span>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-291\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-142.png\" alt=\"\" width=\"384\" height=\"54\" \/>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">1.\u00a0 \u00a0See the F-table (at particular significance level) and find out value of FTab.<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">2. If Fcal &gt; FTab Reject the hypothesis<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Fcal &lt; FTab Accept the hypothesis<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Example:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">After the completion of the training program, the company's statistical staff chose 16 new employees assigned at random to the 3 training methods to study which out of the three training programs is best.<\/span><\/p>\r\n\r\n<div>\r\n<p style=\"text-align: center\">Ans. Table 1. Daily production of 16 new employees<\/p>\r\n<img class=\"aligncenter size-full wp-image-292\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-143.png\" alt=\"\" width=\"403\" height=\"487\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-293\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-144.png\" alt=\"\" width=\"423\" height=\"55\" \/>\r\n\r\n<strong style=\"text-align: initial;text-indent: 1em;font-size: 1em\">HYPOTHESIS<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The three samples could have drawn from H0 : = 2 = 3 Null Hypothesis population having the same mean . Means method of training does not influence the productivity of the employee.<\/span><\/p>\r\n\r\n<\/div>\r\nH1:\u00a0 1, 2 and 3 are not equal\u00a0 alternative hypothesis.\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Step \u2013 1: Calculate the variance among the sample means<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-294\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-145.png\" alt=\"\" width=\"599\" height=\"265\" \/>\r\n\r\n<img class=\"aligncenter size-full wp-image-295\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-146.png\" alt=\"\" width=\"214\" height=\"79\" \/>\r\n\r\n<img class=\"aligncenter size-full wp-image-296\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-147.png\" alt=\"\" width=\"302\" height=\"217\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Step - 2: Calculating the variance within the samples<\/strong>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-297\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-148.png\" alt=\"\" width=\"608\" height=\"416\" \/><img class=\"aligncenter wp-image-298\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-149.png\" alt=\"\" width=\"605\" height=\"286\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-299\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-150.png\" alt=\"\" width=\"357\" height=\"127\" \/>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Step \u2013 3: Compare the 100 estimates of the population variance by reputing their ratio<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-300\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-151.png\" alt=\"\" width=\"396\" height=\"71\" \/>\r\n\r\n<strong>Step \u2013 4: Testing of Hypothesis<\/strong>\r\n\r\n&nbsp;\r\n\r\nCalculate the number of degrees of freedom in the numerator of F ratio.\r\n\r\n&nbsp;\r\n\r\nNumber of degrees of freedom for numerator = (number of samples \u2013 1)\r\n\r\n&nbsp;\r\n\r\n= 3 \u2013 1\r\n\r\n&nbsp;\r\n\r\n= 2\r\n\r\n&nbsp;\r\n\r\nNumber of degrees of freedom for denominator = (nj-1) = nT-K\r\n\r\n&nbsp;\r\n\r\n= (5-1)+(5-1)+(6-1)\r\n\r\n&nbsp;\r\n\r\n= 16 \u2013 3\r\n\r\n&nbsp;\r\n\r\n= 13\r\n\r\n&nbsp;\r\n\r\nWhere nT= total sample size, K=Types of samples\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Now, suppose the director wants to test at 0.05 level the hypothesis, look at table value of F-test or particular numerator value (2) and denominator value (13).<\/p>\r\n&nbsp;\r\n\r\nThe table value of F = 3.81\r\n\r\n&nbsp;\r\n\r\nand since the table value 3.81 sets the upper limit of acceptance and Fcal &lt; Ftab\r\n\r\n&nbsp;\r\n\r\nThe Null hypothesis is accepted.\r\n\r\n&nbsp;\r\n\r\n<strong>7.\u00a0<\/strong><strong>Short-Cut Method<\/strong>\r\n\r\n&nbsp;\r\n\r\nThe values of SSTR and SSE can be calculated by applying the following short-cut methods:\r\n\r\n&nbsp;\r\n\r\nCalculate the grand total of all observations in sample, T\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-301\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-152.png\" alt=\"\" width=\"241\" height=\"41\" \/>\r\n\r\n&nbsp;\r\n\r\nCalculate the correction factor \u00a0;\u00a0<img class=\"aligncenter size-full wp-image-302\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-153.png\" alt=\"\" width=\"211\" height=\"34\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Find the sum of squares of all observations in samples from each of r samples and subtract CF from this sum to obtain the total sum of squares of deviations SST:<\/p>\r\n<img class=\"aligncenter size-full wp-image-303\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-154.png\" alt=\"\" width=\"388\" height=\"148\" \/>\r\n\r\n&nbsp;\r\n<ol start=\"8\">\r\n \t<li style=\"text-align: justify\"><strong>Coding Method: <\/strong>Sometimes the method explained above takes a lot of computational time due to the magnitude of numerical values of observations. The coding method is based on the fact that the F-test statistic used in the analysis of variance is the ratio of variances without unit of measurement. Thus its values does not change if an appropriate constant value is either multiplied, divided, subtracted or added to each of the observations in the sample data. This adjustment reduces the magnitude of numerical values in the sample data and reduces computational time to calculate F value without any change.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n<strong>SUMMARY<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This module provides a statistical test concerning if the means of several groups are all equal and it is a simplest form. ANOVA is equivalent to Student's t-test when only two groups are involved. ANOVA refers to statistical models and associated procedures, in which the observed variance is partitioned into components due to different explanatory variables. If a statistically significant effect is found in ANOVA, one or more tests of appropriate kinds will follow up, in order to assess which groups are different from which other groups or to test various other focused hypothesis.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\"><strong>Learn More:<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li>Sharma, J K (2014), Business Statistics, S Chand &amp; Company, N Delhi.<\/li>\r\n \t<li>Bajpai, N (2010) Business Statistics, Pearson, N Delhi.<\/li>\r\n \t<li>Trevor Hastie, Robert Tibshirani, Jerome Friedman (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition, Springer.<\/li>\r\n \t<li>Darrell Huff (2010), How to Lie with Statistics,\u00a0 W. W. Norton, California.<\/li>\r\n \t<li>K.R. Gupta (2012), Practical Statistics, Atlantic Publishers &amp; Distributors (P) Ltd., N. Delhi.<\/li>\r\n<\/ol>","rendered":"<div>\n<p>&nbsp;<\/p>\n<p><strong>Introduction<\/strong><\/p>\n<ul>\n<li>Experimental Design of Analysis of Variance<\/li>\n<li>Types of Experimental Design and Analysis and Variance<\/li>\n<li>Assumptions for ANOVA<\/li>\n<li>Steps of solving the experimental design<\/li>\n<li>Test for the Equality of <em>k<\/em> Population Means<\/li>\n<li>Test for the Equality of <em>k<\/em> Population Means: An Observational Study<\/li>\n<li>Summary<\/li>\n<li>Self-Check Exercise<\/li>\n<\/ul>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Quadrant-I<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Analysis of Variance and Experimental Design: testing for equality of k population means<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Learning Objectives:<\/strong><\/p>\n<ul>\n<li>After the completion of this module the student will understand:<\/li>\n<li>Experimental Design of Analysis of Variance<\/li>\n<li>Types of Experimental Design and Analysis and Variance Assumptions for ANOVA<\/li>\n<li>Steps of solving the experimental design<\/li>\n<li>Test for the Equality of <em>k<\/em> Population Means<\/li>\n<li>Test for the Equality of <em>k<\/em> Population Means: An Observational Study<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>Introduction of ANOVA<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The analysis of variance frequently referred to by the contraction ANOVA is a statistical technique specially designed to test whether the means of more than TWO quantitative populations are equal.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This technique developed by R.A. FISHER. In 1920 is capable of fruitful application to a diversity of practical problems. Basically it consists of classifying and cross-classifying statistical results and testing whether the means of a specified population differ significantly.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Introduction to Experimental Design and Analysis of Variance<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The statistical studies can be classified as being either experimental or observational. In an experimental study, one or more factors are controlled so that data can be obtained about how the factors influence the variables of interest but as far as in an observational study, no attempt is made to control the factors. One of the important points is that in this technique cause and effect relationship are easier to establish in experimental studies than in observational studies.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Analysis of variance (ANOVA) is used to analyze the data obtained from experimental or observational studies. In this study factor consider as a variable that the experimenter has selected for investigation also a treatment is a level of a factor it can be explained by an example, if location is a factor, and then a treatment of location can be Delhi, Ghaziabad, and Noida. So it concludes that experimental units are the objects of interest in the experiment. A completely randomized design is an experimental design in which the treatments are randomly assigned to the experimental units.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Types of Experimental Design and Analysis and Variance<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Three types of experimental designs are introduced.<\/p>\n<ul>\n<li>A completely randomly design<\/li>\n<li>A randomized block design<\/li>\n<li>A factorial experiment<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>Assumptions for ANOVA<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>For each population, the response (dependent) variable is <em>normally distributed<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The variance of the response variable, denoted <em>s<\/em> 2, is the <em>same<\/em> for all of the populations. The observations must be <em>independent<\/em>.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Steps for calculation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Between-Treatments Estimate of Population Variance <em>s<\/em> <\/strong><strong>2<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The estimate of <em>s<\/em> 2 based on the variation of the sample means is called the mean square due to treatments and is denoted by MSTR.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-285\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-136.png\" alt=\"\" width=\"445\" height=\"158\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-136.png 495w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-136-300x107.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-136-65x23.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-136-225x80.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-136-350x124.png 350w\" sizes=\"auto, (max-width: 445px) 100vw, 445px\" \/><\/p>\n<p>k is the number of treatments (total of samples)<\/p>\n<p><span style=\"font-size: 1em\">n <sub>j<\/sub> is the number of observations in treatment j<\/span><\/p>\n<p>x<sub>j\u00a0\u00a0<\/sub>is\u00a0the sample mean of treatment j<\/p>\n<p><del>x<\/del> is the overall mean, i.e. the average value of ALL the observations from all the treatments<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-286\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-137.png\" alt=\"\" width=\"153\" height=\"157\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-137.png 153w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-137-65x67.png 65w\" sizes=\"auto, (max-width: 153px) 100vw, 153px\" \/><\/p>\n<p><strong>Within-Treatments Estimate of Population Variance <em>s<\/em> <\/strong><strong>2<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The estimate of <em>s<\/em> 2 based on the variation of the sample observations within each sample is called the mean square due to error and is denoted by MSE.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-287\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-138.png\" alt=\"\" width=\"221\" height=\"115\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-138.png 221w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-138-65x34.png 65w\" sizes=\"auto, (max-width: 221px) 100vw, 221px\" \/><\/p>\n<p>Whereas, MSE is mean square due to error<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Test for the Equality of <em>k<\/em> Population Means<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Hypothesis<\/p>\n<p>&nbsp;<\/p>\n<p><em>H<\/em>0 (null hypothesis):<em>\u00a0 m<\/em>1 =<em> m<\/em>2 =<em> m<\/em>3 = . . . =<em> m<\/em>k<\/p>\n<p>&nbsp;<\/p>\n<p><em>H<\/em>a (alternative hypothesis): Not all population means are equal<\/p>\n<p>&nbsp;<\/p>\n<p>Test Statistic<\/p>\n<p>&nbsp;<\/p>\n<p>(MSTR= Mean square due to treatment, MSE= Mean square due to error)<\/p>\n<p>&nbsp;<\/p>\n<p>Rejection Rule<\/p>\n<p>&nbsp;<\/p>\n<p><em>p<\/em>-value Approach:\u00a0 Reject<em> H<\/em>0 if<em> p<\/em>-value &lt;<em> a<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>Critical Value Approach: Reject <em>H<\/em>0 if <em>F<\/em> &gt; <em>F<\/em><em>a<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>Whereas the value of <em>F<\/em><em>a<\/em> is based on an <em>F<\/em> distribution with <em>k<\/em> &#8211; 1 numerator d.f. and <em>n<\/em>T &#8211; <em>k<\/em> denominator d.f.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>6.\u00a0<\/strong><strong>Test for the Equality of <em>k<\/em> Population Means: An Observational Study<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>1.\u00a0 \u00a0Determine one estimate of the population variance from the <strong>&#8216;variance among<\/strong> <strong>the sample means&#8217;<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0 \u00a0Determine a second estimate of the population variance from the <strong>&#8216;variance<\/strong> <strong>within the samples&#8217;<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p>3.\u00a0\u00a0<strong>Compare these two estimates<\/strong>: If they are approximately equal in value, accept the null hypothesis.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-288\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-139.png\" alt=\"\" width=\"374\" height=\"164\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-139.png 374w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-139-300x132.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-139-65x29.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-139-225x99.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-139-350x153.png 350w\" sizes=\"auto, (max-width: 374px) 100vw, 374px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>where <em>x<\/em>\u00a0 = sample mean<\/p>\n<p>x\u00a0\u00a0 = grand mean<\/p>\n<p>k = number of samples<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-290\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-141.png\" alt=\"\" width=\"311\" height=\"44\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-141.png 311w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-141-300x42.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-141-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-141-225x32.png 225w\" sizes=\"auto, (max-width: 311px) 100vw, 311px\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Number of degree of freedom in the numerator of the F ratio = (number of samples &#8211; 1).<\/p>\n<p>Number of degree of freedom in the denomination of the F ratio = (nj\u20131) = nT-K<\/p>\n<p><span style=\"font-size: 1em\">n<\/span><sub>j<\/sub><span style=\"font-size: 1em\"> = size of jth sample n<\/span><sub>T<\/sub><span style=\"font-size: 1em\"> =\u00a0 n<\/span><sub>j<\/sub><span style=\"font-size: 1em\"> = total sample size<\/span><\/p>\n<p><span style=\"font-size: 1em\">k = number of samples<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-291\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-142.png\" alt=\"\" width=\"384\" height=\"54\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-142.png 384w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-142-300x42.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-142-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-142-225x32.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-142-350x49.png 350w\" sizes=\"auto, (max-width: 384px) 100vw, 384px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">1.\u00a0 \u00a0See the F-table (at particular significance level) and find out value of FTab.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">2. If Fcal &gt; FTab Reject the hypothesis<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Fcal &lt; FTab Accept the hypothesis<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Example:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">After the completion of the training program, the company&#8217;s statistical staff chose 16 new employees assigned at random to the 3 training methods to study which out of the three training programs is best.<\/span><\/p>\n<div>\n<p style=\"text-align: center\">Ans. Table 1. Daily production of 16 new employees<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-292\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-143.png\" alt=\"\" width=\"403\" height=\"487\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-143.png 403w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-143-248x300.png 248w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-143-65x79.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-143-225x272.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-143-350x423.png 350w\" sizes=\"auto, (max-width: 403px) 100vw, 403px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-293\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-144.png\" alt=\"\" width=\"423\" height=\"55\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-144.png 423w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-144-300x39.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-144-65x8.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-144-225x29.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-144-350x46.png 350w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\" \/><\/p>\n<p><strong style=\"text-align: initial;text-indent: 1em;font-size: 1em\">HYPOTHESIS<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The three samples could have drawn from H0 : = 2 = 3 Null Hypothesis population having the same mean . Means method of training does not influence the productivity of the employee.<\/span><\/p>\n<\/div>\n<p>H1:\u00a0 1, 2 and 3 are not equal\u00a0 alternative hypothesis.<\/p>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Step \u2013 1: Calculate the variance among the sample means<\/strong><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-294\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-145.png\" alt=\"\" width=\"599\" height=\"265\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-145.png 599w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-145-300x133.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-145-65x29.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-145-225x100.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-145-350x155.png 350w\" sizes=\"auto, (max-width: 599px) 100vw, 599px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-295\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-146.png\" alt=\"\" width=\"214\" height=\"79\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-146.png 214w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-146-65x24.png 65w\" sizes=\"auto, (max-width: 214px) 100vw, 214px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-296\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-147.png\" alt=\"\" width=\"302\" height=\"217\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-147.png 302w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-147-300x216.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-147-65x47.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-147-225x162.png 225w\" sizes=\"auto, (max-width: 302px) 100vw, 302px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Step &#8211; 2: Calculating the variance within the samples<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-297\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-148.png\" alt=\"\" width=\"608\" height=\"416\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-148.png 608w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-148-300x205.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-148-65x44.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-148-225x154.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-148-350x239.png 350w\" sizes=\"auto, (max-width: 608px) 100vw, 608px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-298\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-149.png\" alt=\"\" width=\"605\" height=\"286\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-149.png 605w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-149-300x142.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-149-65x31.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-149-225x106.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-149-350x165.png 350w\" sizes=\"auto, (max-width: 605px) 100vw, 605px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-299\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-150.png\" alt=\"\" width=\"357\" height=\"127\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-150.png 357w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-150-300x107.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-150-65x23.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-150-225x80.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-150-350x125.png 350w\" sizes=\"auto, (max-width: 357px) 100vw, 357px\" \/><\/p>\n<\/div>\n<div><\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Step \u2013 3: Compare the 100 estimates of the population variance by reputing their ratio<\/strong><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-300\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-151.png\" alt=\"\" width=\"396\" height=\"71\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-151.png 396w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-151-300x54.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-151-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-151-225x40.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-151-350x63.png 350w\" sizes=\"auto, (max-width: 396px) 100vw, 396px\" \/><\/p>\n<p><strong>Step \u2013 4: Testing of Hypothesis<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Calculate the number of degrees of freedom in the numerator of F ratio.<\/p>\n<p>&nbsp;<\/p>\n<p>Number of degrees of freedom for numerator = (number of samples \u2013 1)<\/p>\n<p>&nbsp;<\/p>\n<p>= 3 \u2013 1<\/p>\n<p>&nbsp;<\/p>\n<p>= 2<\/p>\n<p>&nbsp;<\/p>\n<p>Number of degrees of freedom for denominator = (nj-1) = nT-K<\/p>\n<p>&nbsp;<\/p>\n<p>= (5-1)+(5-1)+(6-1)<\/p>\n<p>&nbsp;<\/p>\n<p>= 16 \u2013 3<\/p>\n<p>&nbsp;<\/p>\n<p>= 13<\/p>\n<p>&nbsp;<\/p>\n<p>Where nT= total sample size, K=Types of samples<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Now, suppose the director wants to test at 0.05 level the hypothesis, look at table value of F-test or particular numerator value (2) and denominator value (13).<\/p>\n<p>&nbsp;<\/p>\n<p>The table value of F = 3.81<\/p>\n<p>&nbsp;<\/p>\n<p>and since the table value 3.81 sets the upper limit of acceptance and Fcal &lt; Ftab<\/p>\n<p>&nbsp;<\/p>\n<p>The Null hypothesis is accepted.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>7.\u00a0<\/strong><strong>Short-Cut Method<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>The values of SSTR and SSE can be calculated by applying the following short-cut methods:<\/p>\n<p>&nbsp;<\/p>\n<p>Calculate the grand total of all observations in sample, T<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-301\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-152.png\" alt=\"\" width=\"241\" height=\"41\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-152.png 241w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-152-65x11.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-152-225x38.png 225w\" sizes=\"auto, (max-width: 241px) 100vw, 241px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Calculate the correction factor \u00a0;\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-302\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-153.png\" alt=\"\" width=\"211\" height=\"34\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-153.png 211w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-153-65x10.png 65w\" sizes=\"auto, (max-width: 211px) 100vw, 211px\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Find the sum of squares of all observations in samples from each of r samples and subtract CF from this sum to obtain the total sum of squares of deviations SST:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-303\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-154.png\" alt=\"\" width=\"388\" height=\"148\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-154.png 388w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-154-300x114.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-154-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-154-225x86.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-154-350x134.png 350w\" sizes=\"auto, (max-width: 388px) 100vw, 388px\" \/><\/p>\n<p>&nbsp;<\/p>\n<ol start=\"8\">\n<li style=\"text-align: justify\"><strong>Coding Method: <\/strong>Sometimes the method explained above takes a lot of computational time due to the magnitude of numerical values of observations. The coding method is based on the fact that the F-test statistic used in the analysis of variance is the ratio of variances without unit of measurement. Thus its values does not change if an appropriate constant value is either multiplied, divided, subtracted or added to each of the observations in the sample data. This adjustment reduces the magnitude of numerical values in the sample data and reduces computational time to calculate F value without any change.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p><strong>SUMMARY<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This module provides a statistical test concerning if the means of several groups are all equal and it is a simplest form. ANOVA is equivalent to Student&#8217;s t-test when only two groups are involved. ANOVA refers to statistical models and associated procedures, in which the observed variance is partitioned into components due to different explanatory variables. If a statistically significant effect is found in ANOVA, one or more tests of appropriate kinds will follow up, in order to assess which groups are different from which other groups or to test various other focused hypothesis.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><strong>Learn More:<\/strong><\/p>\n<ol>\n<li>Sharma, J K (2014), Business Statistics, S Chand &amp; Company, N Delhi.<\/li>\n<li>Bajpai, N (2010) Business Statistics, Pearson, N Delhi.<\/li>\n<li>Trevor Hastie, Robert Tibshirani, Jerome Friedman (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition, Springer.<\/li>\n<li>Darrell Huff (2010), How to Lie with Statistics,\u00a0 W. W. Norton, California.<\/li>\n<li>K.R. Gupta (2012), Practical Statistics, Atlantic Publishers &amp; Distributors (P) Ltd., N. Delhi.<\/li>\n<\/ol>\n","protected":false},"author":3,"menu_order":27,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-pankaj-madan"],"pb_section_license":""},"chapter-type":[],"contributor":[61],"license":[],"class_list":["post-281","chapter","type-chapter","status-publish","hentry","contributor-prof-pankaj-madan"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/281","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/281\/revisions"}],"predecessor-version":[{"id":305,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/281\/revisions\/305"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/281\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/media?parent=281"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapter-type?post=281"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/contributor?post=281"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/license?post=281"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}