Question

How do you calculate the sample mean of the differences?

How do you know which inequality symbol (less than or greater than) to use when setting up your claim? (I.e. When do you test if the difference between the matched pair data is less than zero? When is the difference greater than zero?

Which distribution is used for testing a claim about dependent samples?

Answer 1

Suppose, X1,....,Xn₁ =Sample 1 of size n₁

Y1,.....,Yn₂ = Sample 2 of size n₂

Formula for sample mean of the differences :
= ∑x_i/n

= ∑y_i/n

Sample mean of differences : -

So while hypothesis testing, we usually put the equal symbol in the null hypothesis and whatever the researcher claims or assumes in the alternative hypothesis, it can be anything like less than, or greater than, or unequal.
Suppose the researcher claims than mean of X is greater than that of Y, in that case we go for alternative hypothesis H1 : mu_x > mu_y or H1 : difference_(x-y) > 0. Suppose he claims mean of X is lesser than that of Y, we go for H1 : mu_x < mu_y or H1 : difference_(x-y) < 0. But if you take difference_(y-x) then the whole thing will get reversed.

Bivariate normal distribution is assumed in matched pair data or dependent samples data so that the difference X - Y reduces to marginal normal distribution and we can carry out the t-test.