Question

Statistics:

For a two-sample t-test, where the 2 population variances are assumed to be unequal, how can I show that the statistic \[ \large \frac{(\overline{X}-\overline{Y})-(\mu_1-\mu_2)}{\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}\sim t(\nu)\] is approximately t-distributed with degrees of freedom: \[ \Large \nu=\frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2} \right)^2}{\frac{\left(\frac{s_1^2}{n_1} \right)^2}{n_1-1}+\frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}\]

Answer 1

v = n-1

Answer 2

Here: \( \large\overline{X}=\frac{1}{n_1}\sum\limits_{i=1}^{n_1}X_i\) where \(X_1, \ldots, X_{n_1}\) are i.i.d. \(\text{N}\left(\mu_1, \sigma_1^2 \right)\), \( \large \overline{Y}=\frac{1}{n_2}\sum\limits_{i=1}^{n_2}Y_i\) where \(Y_1, \ldots, Y_{n_2}\) are i.i.d. \(\text{N}\left(\mu_2, \sigma_2^2 \right)\), \(S_1^2\) and \(S_2^2\) are the unbiased sample variances for \(X_i\) and \(Y_i\) respectively

Answer 3

\(\nu = n-1\) if we have a one-sample t-test

Answer 4

Looks like you're basically establishing this equation: http://en.wikipedia.org/wiki/Welch%E2%80%93Satterthwaite_equation

Answer 5

Here's a link that looks like it explains it. I haven't given it the attention it needs to fully understand what's going on though: http://www.public.iastate.edu/~dnett/S511/21CochranSatterthwaite.pdf

Answer 6

That was very helpful. Knowing this too has led me to other resources too . I was kind of going nowhere before in my searches xD

Answer 7

Happy to help :)