The usual t-statistic for testing a difference between the means of two independently sampled groups, under the assumptions of normality and equal group variances, is Here, Y 1 and Y 2 are the means...

The usual t-statistic for testing a difference between the means of two independently sampled groups, under the assumptions of normality and equal group variances, is

Here, Y₁
and Y₂
are the means of the two groups, n₁
and n₂
are the numbers of observations in the groups, and Y_i1
and Y_i2
are the observations themselves. Let F₀
be the one-way ANOVA F-statistic for testing the null hypothesis H₀: µ₁
= µ₂. Prove that t
²
₀
= F₀
and that, consequently, the two tests are equivalent.