Question

"Y's being independent. In what follows assume that ? and that ?1 and ?, are both unknown. Consider the hypotheses Ho : ??-ly-do and Hi : ??-Pr, * do.

what is the distribution of the test statistics when Null(Ho) is true?

Answer 1

When H0 is true,

$\mu_x - \mu_y$ follows Normal Distribution, as linear combination of normally distributed variables follows normal distribution.

Mean = $\mu_x - \mu_y = d_0$

Variance: (Pooled variance when individual variances are equal but unknown)

$\sigma^2 = \frac{(n_X-1)s_X^2 + (n_Y-1)s_Y^2 + }{n_X + n_Y-2}$

where s denotes the sample standard deviation for sample X and Y.