Question

Explain why the critical value increases as k (degrees of freedom) increases?

Answer 1

The critical value would be determined using df for t distribution. Note as df increases the t distribution tends to the normal distribution.

Now see the formula for t is,

(sample mean - population mean)/sample s.d/√k

So as k increases the value of t would also increase. Degrees of freedom for the test is k-1. So obviously as k-1 increases k would increase.

Now as k increases or approaches infinity it follows a normal distribution so the critical value increases.