PS296 Chapter Notes - Chapter 14: Standard Deviation, Variance, Weighted Arithmetic Mean
Pooling Variances:
-the above equation for t is only relevant when the sample sizes are equal
-one of the assumptions required for the use of t for independent samples is that σ²1 = σ²2,
regardless of the truth or falsity of H0 (homogeneity of variance rule) and because the pop.
Variances are assumed to be equal, this common variance can be represented by σ² with no
subscript
-however with two independent samples, we have 2 estimates of σ² (s²1 and s²2) and thus we
need to obtain an average of the two that is a better estimate σ² than either of the two separate
estimates
-we want a weighted average in which the sample variances are weighted by their degrees of
freedom (ni – 1)
New estimate called s²p:
-numerator represents the sum of the variances, each weighted by its degrees of freedom
-denominator is the sum of the weights/degrees of freedom for s²p
-the weighted average of the two sample variances is referred to as a pooled variance estimate
We can now replace s²i in the formula for t with s²p to get:
Document Summary
The above equation for t is only relevant when the sample sizes are equal. One of the assumptions required for the use of t for independent samples is that 1 = 2, regardless of the truth or falsity of h0 (homogeneity of variance rule) and because the pop. Variances are assumed to be equal, this common variance can be represented by with no subscript. However with two independent samples, we have 2 estimates of (s 1 and s 2) and thus we need to obtain an average of the two that is a better estimate than either of the two separate estimates. We want a weighted average in which the sample variances are weighted by their degrees of freedom (ni 1) Numerator represents the sum of the variances, each weighted by its degrees of freedom. Denominator is the sum of the weights/degrees of freedom for s p.
