How Does Population Size Impact Required Sample Size in Statistical Testing?

[tedpark1212]

So I made an attempt to solve question 1, but I am stuck with question 2. I know that I need to use binominal distribution, but am unclear of the proper steps. Can anyone provide some help?

1. Suppose you want to show the significance of an effect size of 0.20 between your sample mean and a hypothesized mean. You intend to conduct a two-sided one-sample t-test. The sample variance is σ2 = 1.0. You assume that the population is large, or effectively infinite. If you want the test to have a power of 0.80, what is the required sample size for the following?

Attempt to solve:

• A level of significance of α = 0.05

n = 2 σ2 (Z (beta) + Z (alpa/2)) 2/ effect size2
n = 2(1) ( 1.28 + 1.96) 2/ 0.202------ for α = 0.05
n = 2 (20.9952) / 0.04
n=524.88
n= 525

The required sample size is 525 with a level of significance of α = 0.05.
• A level of significance of α = 0.01
n = 2 σ2 ( Z (beta) + Z (alpha/2) ) 2/ effect size2
n = 2(1) ( 1.28 + 2.575) 2/ 0.202------ for α = 0.01
n= 2 (14.86) / 0.04
n=743.05
n= 743
The required sample size is 743 with a level of significance of α = 0.01.

2. After some research, you determine that the size of the population is N = 350. If you want the test to have a power of 0.80, what is the required sample size for the following?

• A level of significance of α = 0.05• A level of significance of α = 0.013. Comment on the sample sizes required for questions 1 and 2. Explain the differences and similarities.

 

[lippyka]

The sample size required for question 1 is much larger than the sample size required for question 2, because in question 1 we assumed that the population is effectively infinite. This means that the sample size can be increased without any concern for the size of the population. In question 2, however, the population size is known to be 350, so the sample size must be smaller due to the finite size of the population. Despite this difference, both questions require an increase in sample size when the level of significance is decreased (i.e. when alpha is decreased from 0.05 to 0.01). This is because a lower level of significance requires more precision in the estimation of the true mean, which can only be achieved with a larger sample size.