Calculating Power for a Study: Population µ = 100, σ = 20

[TrielaM]

I have had some issues with an equation in one of my classes, hoping some one can point out what I am going wrong. If anyone can point out some flaw in my process or calculations I would appreciate it. The problem lists:

Population µ = 100, σ = 20. Planning to have a sample of 50 participants, and expect the sample mean to be 5 points different from the population mean. Use non-directional hypothesis and α = .05.

What is the Power for this planned study?​

My calculations:
σ /(SqrRt)n = 20/(SqrRt) 50 = 20/7.071 = 2.828 Standard error
α = .05 = z = +1.96
1.96(2.828) = 5.543

z= (M-µ)/Standard Error = (105.543-105)/2.828 = .543/2.828 = 0.19

Z Table: .19 = .5753
Power = 57.53%

 

[I like Serena]

I have had some issues with an equation in one of my classes, hoping some one can point out what I am going wrong. If anyone can point out some flaw in my process or calculations I would appreciate it. The problem lists:

Population µ = 100, σ = 20. Planning to have a sample of 50 participants, and expect the sample mean to be 5 points different from the population mean. Use non-directional hypothesis and α = .05.

What is the Power for this planned study?​

My calculations:
σ /(SqrRt)n = 20/(SqrRt) 50 = 20/7.071 = 2.828 Standard error
α = .05 = z = +1.96
1.96(2.828) = 5.543

z= (M-µ)/Standard Error = (105.543-105)/2.828 = .543/2.828 = 0.19

Z Table: .19 = .5753
Power = 57.53%

Welcome to MHB, TrielaM! :)

It appears you have calculated the type II error $\beta$.
The power is its complement.

In your case:
\begin{aligned}\beta &= 57.53\% \\
\text{Power} &= 42.47\% \end{aligned}

 

[TrielaM]

Thank you very much for the reply. So in the process I used I needed to add a step for 1-B. Would this step be applicable all the time with this form of equation or no? The books I was using didn't include this but it was certainly the correct answer, instead it had you pull from a Z table in the section for the body, where as here the answer was found in the tail. At least I know where I was flubbing up though I am still not sure how to make the distinction.

 

[I like Serena]

Thank you very much for the reply. So in the process I used I needed to add a step for 1-B. Would this step be applicable all the time with this form of equation or no?

Yes.
I would suggest always drawing a picture though, like the one I have included in my previous post.

The books I was using didn't include this but it was certainly the correct answer, instead it had you pull from a Z table in the section for the body, where as here the answer was found in the tail. At least I know where I was flubbing up though I am still not sure how to make the distinction.

Apparently you pulled your result from a Z table that gives the area to the left, which is pretty standard and often shown in a small graph next to the table.
As a result your value of z=0.19 gives you the green area.

But you need the blue area, which is the area to the right of z=0.19.

Note that the power of a test (the blue area) is the probability that we reject the null hypothesis when it should indeed be rejected, because the alternative hypothesis is true.

 

[blue_raver22]

Your calculations are incorrect. Here is the correct process to calculate power for this study:

Step 1: Determine the effect size, which is the difference between the population mean and the expected sample mean. In this case, the effect size is 5.

Step 2: Determine the standard deviation of the sampling distribution, which is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard deviation of the sampling distribution is 20/√50 = 2.828.

Step 3: Determine the critical z-score for a non-directional hypothesis with a significance level of α = .05. This can be found using a z-table or a statistical software, and in this case, the critical z-score is 1.96.

Step 4: Calculate the z-score for the effect size. This can be done by dividing the effect size by the standard deviation of the sampling distribution. In this case, the z-score is 5/2.828 = 1.769.

Step 5: Determine the area under the normal curve from the critical z-score to the z-score for the effect size. This can also be found using a z-table or a statistical software, and in this case, the area is 0.0385.

Step 6: Calculate the power by subtracting the area from 1. In this case, the power is 1 - 0.0385 = 0.9615 or 96.15%.

Therefore, the power for this planned study is 96.15%. It is important to note that this is an estimated power and the actual power may differ depending on the sample size, effect size, and other factors.