How to Calculate a Sample Size Given Standard Deviation, Confidence Interval and Margin of Error

The equation we want to use is: Sample Size = (Z*σ / Margin of Error)^2

We’ll use the following example question from the ASQ Black Belt Exam.

Question: When σ = 10, what sample size is needed to specify a 95% confidence interval of ±3 units from the mean?

(A) 7
(B) 11
(C) 32
(D) 43

Answer: 43.  This is a compound question. First we must find the Z score for this confidence interval, then we want to calculate the sample size for that margin of error.

Step 1: Find the Z score

We need Z(α/2), where α is the confidence interval.

You would just look that up on the Z table.

In this case:

α = 95

Z(α/2) = Z(95/2) = Z(47.5) = 1.96

Step 2: Apply the Equation Sample Size = (Z*σ / Margin of Error)^2

Just a simple plug and play equation:

Sample Size = (Z*σ / Margin of Error)^2 = (1.96 * 10 / 3)^2 = (19.6/3)^2 = 6.53^2 = 42.7

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