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(Harder question): Show that the two sample t-test statistic squared is equal to the single-factor ANOVA F statistic when there are two levels. For simplicity, assume equal sample size of n in each of the two groups RECALL: Two sided, two independent sample pooled t-test tests the following hypotheses: The test statistic is: VEA where yi and ỹ2 are the sample means in group 1 and 2, respectively. The pooled variance is given by where s and s3 are the sample variances in the group 1 and 2 respectively-En u

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The t-test with two groups presumes that both groups are nomally distributed with the equal variance although the means may vary under the different hypothesis. That is equal to a regression with a dummy variable as the regression permits the mean of every group to differ except not the variance. Therefore the residuals equal to the data with the group means subtracted contain the similar distribution, that is, they are usually distributed with zero mean A t-test with uneven variances is not equal to a one-way ANOVA The t-test is often thought of as associate extension of the z-test, appropriate for things once the variance of the population isnt better-known and has got to be calculable from the sample variance. The worth oft is then calculated because the distinction between 2 samples suggests that divided by the calculable pooled sample variance in the case of two free lance samples, drawn from populations of equal variance. This can be specifically analogous to the z-test, that uses the distinction within the sample suggests that divided by the better known variance.

However theres in a different way of gazing the t-statistic, that has the advantage of linking the t-test to the following most complicated applied math check, that of the unidirectional analysis of variance. its usually noted that once analysis of variance is applied to merely 2 teams, associated once thus one will calculate each a t-statistic and an F-statistic from the ident ical knowledge, it happens that the 2 are connected by the easy formula: t2-F. This remark is rarely verified in statistics courses, nor even are any heuristic reasons are given why it ought to be therefore. This will leave some confusion within the minds of scholars. Clearly theres some associat ion between the t-test and two-group, unidirect ional analysis of variance, however whats it? The t-statistic is some things resembling a z-vahie, and therefore the F-statistic is actually the quantitative relation of 2 estimates of variance: why ought t<o there be any relationship between the two? One way to appear at this can be to think about the t-statistic because the quantitative relation of 2 estimates of ordinary deviat ion, that yields the sq. of t as being the quantitative relation of 2 estimates of variance. truly but its a lot of natural to come back at it the opposite means spherical associated to write down an expression for the quantitative relation of 2 variance estimates for a population, associated derive an expression from it that then is seen to be similar to the same old formula for t2. This approach even works for the powerful case of various population customary deviations, i.e. the Behrens-Fisher state of affairs; therefore its some power as a heuristic approach.

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