1. The F-test for Equality of Two Variances

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1 . The F-tet for Equality of Two Variance Previouly we've learned how to tet whether two population mean are equal, uing data from two independent ample. We can alo tet whether two population variance are equal uing ample data. The F hypothei tet i defined a: σ σ H 0 : = H : σ σ Tet Statitic: F F = if = if < where and are the ample variance. The more thi ratio exceed from, the tronger the evidence for unequal population variance. The tatitical ignificance of F i found by integrating an area of a cumulative F ditribution. A with the t- and χ ditribution, we have a different F ditribution according to our degree of freedom. However for the F tatitic, we mut conider the df aociated with both variance etimate, i.e., (df, df ) degree of freedom. Figure. F Ditribution for Variou (df, df )

2 In theory, the F-ditribution i two-tailed. That i, if F < we can integrate the ditribution from 0 to F, or if F >, from F to +inf. However (poibly due to conideration of numerical accuracy) the convention i to alway integrate the poitive tail. For thi reaon we alway place the larger of the two variance etimate in the numerator, and chooe F ditribution with: Excel (n, n ) df when, or (n, n ) df when <. In Excel we can calculate F a the ratio ample variance, and then ue the FDIST function to compute the p-value of F. Where: = FDIST(F, df, df) The larger variance hould alway be placed in the numerator. Multiply reult for a two-tailed tet. To enure the larger variance i alway in the numerator, we ue conditional logic in the cell formula for F: Conditional logic in Excel: =IF(logical expreion, formula if TRUE, formula if FALSE) So, formula for F: =IF(var>var, var/var, var/var)

3 Aumption of F tet of variance: In the population from which the ample were obtained the variable i normally ditributed.. Credible and Confidence Interval You will recall that we've often made the ditinction between credible interval and confidence interval. Credible interval are derived uing Bayeian tatitic, and confidence interval uing claical tatitical method. Alo recall that the credible interval ha a very clear and ueful definition: it i the expected or ditribution of ome population parameter of, baed on oberved ample data. For example the 95% credible interval of a mean would tell u the etimated range for a population mean. Again recall that the confidence interval, on the other hand, ha a very convoluted and confuing definition, one not terribly helpful. In fact, the real reaon people ue confidence interval i becaue they tend give them an interpretation that actually applie to the credible interval. Fortunately, in everal common cae, uch a the z- and t-tatitic, the confidence interval i, given mild aumption, exactly equal to the credible interval. Here we will how why. Suppoe we have a random ample of n cae. We can compute the ample mean and ample tandard deviation, and from thee contruct an expected ampling ditribution. The mean of our ampling ditribution would be our ample mean, and the tandard deviation would be / n.

4 Suppoe, for convenience, that our ample mean i exactly 0, and our tandard error i exactly. Now uppoe we wih to etimate, given thi ampling ditribution, the probability (actually a probability denity) of drawing a new ample of exactly n cae that ha a ample mean of -. The uual formula for the probability denity of a normal ditribution i: But becaue our mean i 0 and tandard deviation i thi implifie to: f ( x) = e π.5 z If z =, the above formula give a value of 0.4. Here' where the idea of "if I'm x unit away from you, you're alo x unit away from me" applie. Suppoe now that our population mean actually were -, then our actual mean, 0, would correpond to a z of. And uing the ame formula above, the denity of thi value would alo be 0.4. Further, we could follow the ame arrangement for every poible value for the population mean of our ampling ditribution. In other word, in every cae: P( = c µ = c ) = P( µ = c = c ) = P( µ = c c ) = Where c and c are any two value. The only requirement for thi to work i the aumption that the tandard error, for every poible value ofµ. Thi i true for both the z and the t ditribution., i the ame The aumption of a contant tandard error of the ampling ditribution, however, i not true in the cae of a ample proportion. Recall that there the tandard error i etimated a: ( p )( p) N So a p change, o doe the tandard error of p. Therefore we need another approach to contruct the credible interval of a proportion. 3. Credible Interval for a Proportion

5 Cla demontration