RVUC- Statistics Short Note (2).doc

If each individual test were conducted using a level

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If each individual test were conducted using a level of significance of say α = 0.05, then the overall level of significance would be higher than 0.05. For example, if Ho: μ 1 = μ 2 = μ 3 , α (the probability of rejecting a true null hypothesis) = 0.143 (1-0.95 3 ). Thus, we want to test simultaneously for differences among the means of all the populations, and we want the joint level of significance of the test to be α. To perform this test we make use of the F-distribution and use a method called ANOVA. In order to use ANOVA, we assume the following: All the samples were randomly selected and are independent of one another. The populations from which the samples were drawn are normally distributed. If however, the sample sizes are large enough, we do not need the assumption of normality. All the population variances are equal. ANOVA is based on a comparison of two different estimates of the variances, σ 2 , of overall population. 1. The variance obtained by calculating the variation within the samples themselves – Mean Square within (MSW). 2. The variance obtained by calculating the variation among sample means – Mean Square between (MSB). Since both are estimates of σ 2 , they should be approximately equal in value when the null hypothesis is true. If the null hypothesis is not true, these two estimates will differ considerably. The three steps in ANOVA, then, are: 23
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24 | P a g e 1. Determine one estimate of the population variance from the variation among sample means 2. Determine a second estimate of the population variance from the variation within the samples 3. Compare these two estimates. If they are approximately equal in value, accept the null hypothesis. Calculating the Variance among the Sample Means (MSB) The variance among the sample means is called Between Column Variance or Mean Square between (MSB). Sample variance = . 1 2 2 n X X S Now, because we are working with sample means and the grand mean, let’s substitute X for X, X for X , and K (number of samples) for n to get the formula for the variance among the sample means: . 1 2 2 K X X S means sample among Variance X In sampling distribution of the mean we have calculated the standard error of the mean as n X . Cross multiplying the terms n X . Squaring both sides n X 2 2 . In ANOVA, we do not have all the information needed to use the above equation to find σ 2 . Specifically, we do not know 2 X . We could, however, calculate the variance among the sample means, 2 X S , using . 1 2 2 K X X S X So, why not substitute 2 X S for 2 X and calculate an estimate of the population variance? This will give us: . , ...... , 1 1 * 2 1 2 2 2 2 equal are n n n If K X X n K X X n n S k X x Which sample size to use? 24
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25 | P a g e There is a slight difficulty in using this equation as it stands. n represents the sample size, but which sample size should we use when different samples have different sizes? We solve this problem by multiplying 2 X X j by its won appropriate n j , and hence 2 X becomes: MSB = 1 2 2
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