Struggling with Statistical Sample Size Calculations?

[tedpark1212]

Can anyone help me check my work for #4? I had trouble solving #4 but made an attempt. Please advise.

3. 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?

• A level of significance of α = 0.05

n = 2 σ2 (Z (beta) + Z (alpha/2)) 2/ effect size2
z(beta) = 1.28, z(alpha) = 1.96 for two-sided test at a level of significance of 0.05
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 for a level of significance of α = 0.05 is 525.
• A level of significance of α = 0.01
n = 2 σ2 ( Z (beta) + Z (alpha/2) ) 2/ effect size2
z(beta) = 1.28, z(alpha) = 2.575 at a level of significance of 0.01
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 for a level of significance of α = 0.01 is 743.

4. 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
n = σ2 (Qe-1 + Qc-1 ) (Z(alpha) + Z(beta) 2 / μ₁²
n = 1(4) (1.645 + 1.282) 2 / (0.20)2
n = 856.7
n= 857
The required sample size for a level of significance of α = 0.05 is 857.
A level of significance of α = 0.01
n = σ2 (Qe-1 + Qc-1 ) (Z(alpha) + Z(beta) 2 / μ₁²
n = 1(4) (2.33 + 1.282) 2 / (0.20) 2
n= 1304.6
n= 1305
The required sample size for a level of significance of α = 0.01 is 1305.

 

[mighty2000]

Hello,

Thank you for sharing your work for #4. Overall, your calculations and approach seem correct. However, there are a few things I would like to point out for clarification.

First, for the calculation of the required sample size for a two-sided t-test, the formula you used is typically used for a one-sided test. For a two-sided test, the formula is n = (2 * σ² * (Z(α/2) + Z(β))²) / d², where d is the effect size. So for this problem, the formula would be n = (2 * 1 * (1.96 + 1.28)²) / 0.20² = 525. You also used the correct values for the z-scores (1.96 for a level of significance of α = 0.05, and 1.28 for a power of 0.80). Therefore, your final answer of 525 is correct for a level of significance of α = 0.05.

Second, for a level of significance of α = 0.01, the z-score for a two-sided test would be 2.575, not 2.33. So the correct calculation would be n = (2 * 1 * (2.575 + 1.28)²) / 0.20² = 743, which is the same answer you got.

Lastly, for the calculation of the required sample size for a two-sided test with a known population size, the formula you used is correct. However, the value for Qe-1 should be (N-n)/(N-1), where N is the population size and n is the sample size. So for this problem, it would be (350-1)/(350-1) = 0.997. Using this value, your final answers for both α = 0.05 and α = 0.01 are correct.

I hope this helps. Keep up the good work!