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Question

A pollster wishes to estimate, with 95% confidence, the proportion of people who approve of how President Trump is performing his job.

Find the minimum sample size necessary to achieve a margin of error of less than 3%.

1. Assuming no previous estimates are available. Round your answer up to the next integer.

1. Assuming a previous estimate of 0.65. Round your answer up to the next integer.

Answer 1

Good afternoon and thank you for the question:

The answers are:

1,068 people assuming no previous estimates

972 people assuming a previous estimate of phat=0.65

Below is an explanation of how I arrived at the above answers:

The equation for the margin of error, MOE, for proportions is:

MOE = z*√phat*(1-phat)/n.

In the above equation, phat is the point estimate for the proportion of people who approve of how President Trump is performing his job, z comes from the standard normal distribution and is equal to 1.96 when we desire 95% confidence in our estimate, phat, and n is the desired sample size.

Rearranging the above equation to solve for n, we get:

n = phat*(1-phat)*z²/MOE²

Because we have no prior point estimate, we want the largest possible sample size, so we assume phat=0.5.

Note: If you play around with different values of phat, you will find that phat=0.5 gives the largest value to phat*(1-phat).

We plug in our knowns to get:

n = 0.5*(0.5)*1.96²/0.03² ≈ 1,067.11. When dealing with sample sizes, we always round up to the nearest integer, so we need 1,068 people in our sample size. This is the smallest sample size that will achieve a margin of error equal to or less than 3%.

Likewise, if we now have a prior estimate of phat=0.65 and we still want 95% confidence and a margin of error less than 3%, we have:

n = 0.65*(1-0.65)*1.96²/0.03² ≈ 971.07, so we need a sample size of n = 972 people.