Question

Looking at the power formula given in class, which of the following statement below is False?

		A-If the effect size increases, we need a smaller sample size
		B-If the sample size increases, power will decrease.
		C-If type II error increases, power will decrease.
		D-If the variance of the sample is large, power will decrease.

Answer 1

A is true

Sample size is given by :

1-a/2 1-8 ES

where ES is the effect size. As effect size increases we need a smaller sample size/

$\dpi{200} {\color{DarkRed} B\text \space is \text \space FALSE}$

Sample size needed typically increases at an increasing rate as power increases.

Conceptual knowledge

The formula for power is Power=1−β

where β is type II error.

and the probability of making a type II error (β) is computed as follows:

$\beta = \Pr( \text{Fail to Reject a False Null Hypothesis Ho} )$

$= \Pr( -z_c \le \frac{\bar X - \mu_0}{\sigma/\sqrt{n}} \le z_c | \mu)$

effect size is

$d= \frac{ \mu_0 - \mu }{ \sigma }$

So β would decrease if sample size increases, thus power would increase. So if sample size increases power will increase.