Question

It is known that the population variance (σ2) is 125. at 95% confidence, what sample size should be taken so that the margin of error does not exceed 3?

Answer 1

To achieve a margin of error of 3 at a 95% confidence level with a population variance of 125, a sample size of approximately 6 is needed.

Step-by-step explanation:

The question posed regards finding an appropriate sample size for a statistical study, where the known population variance (σ62) is 125, and the desired margin of error at a 95% confidence level should not exceed 3. To determine this, we use the formula for the margin of error in relation to the sample mean:

E = z * (σ/√n)

Where E is the margin of error, z is the z-score corresponding to the confidence level, σ is the population standard deviation, and n is the sample size. For a 95% confidence interval, the z-score is typically 1.96. The population standard deviation (σ) can be found by taking the square root of the variance, which is √125 = 11.1803. We set the formula like this:

3 = 1.96 * (11.1803/√n)

And solve for n, which would give us:

n = (1.96 * 11.1803/3)2 = (7.193/3)2 = (2.3977)2 ≈ 5.748, rounded up to 6. Therefore, a sample size of 6 is needed to achieve the desired margin of error.