Aim: To calculate statistical power with given Sample Size (Z test): Difference between a sample proportion and a known / hypothetical proportion.

Formula Used

N = Sample Size in first group = (N1).

P₁ = Proportion of successful outcoe in sample = S₁ / N₁ (Out of 1) (e .g. 25% = 0.25).

P₂ = Known / hypothetical proportion against which P₁ is to be compared (Out of 1) (e .g. 25% = 0.25).

Z _1-α = the standard normal deviate corresponding the confidence level

Φ(x)=P(Z ≤ x). It is the cumulative distribution function (CDF) of normal distribution. Simply, it is the area of the standard normal curve towards left side of x.

Example:

An investigator wanted to conduct a study to test whether proportions of successful examinees in an institute were significantly different than 75%. He randomly selected 200 examinees from the institute, and found that 160 were successful. How much was the power to detect the significant difference between these two proortions, at confidence level of 95 % (alpha = 5%)?

Solution:

Here

S₁ = 160, N₁ = 200, P₂ = 75%, confidence level = 95%, N₁=200, tails = 2 ("significantly different")

After putting these values, we get power = 42.38%