Determining The Minimum Sample Size For Estimating A Population Proportions

In this video we discuss how to calculate or determine the minimum sample size needed to construct a confidence interval for a population proportion. We cover using the margin of error formula and setting it equal to the sample size.

Transcript/notes
Minimum sample size needed for a interval estimate for a population proportion.

When calculating an interval estimate for a population proportion, we use the formula p hat minus z alpha over 2, times the square root of the quantity p hat times q hat, divided by n, less than p, less than p hat plus z alpha over 2, times the square root of the quantity p hat times q hat, divided by n.

And in this formula, z alpha over 2, times the square root of the quantity p hat times q hat, divided by n is the margin of error, which we can label as big E.

N, or the sample size is important in statistics, and we can do some math and set this formula equal to n. In which we get n equals, p hat times q hat times the quantity z alpha over 2 divided by big E, squared.

And in this formula, n equals the sample size, p hat is equal to the proportion of a sample, for example 78% of dogs bark when the doorbell rings, so, in this case, p hat would be equal to 0.78, q hat is equal to 1 minus p hat, so in our dog example q hat would be equal to 1 minus 0.78, which is 0.22, z alpha over 2 is a z score based on the confidence level we are using, which is usually 90%, 95% or 99% (show 3 little graphs for these) and I have the z values listed for those percentages, and big E is the margin of error.

And now we have a formula that tells us the minimum sample size needed for a desired margin of error. 2 quick notes, if a value of p hat is known, say from a previous study, that value can be used in the study, and if no approximation of p hat is known, you should use p hat equal to 0.5, as this gives the largest possible value for p hat times q hat, which gives the largest possible value for n.

Time for an example.
Let’s say that you want to estimate, with 90% confidence, the proportion of people that get married outdoors. A previous study showed that 18% of people had outdoor weddings. And lets say that you want to be accurate within 3% of the true proportion. What is the minimum sample size necessary?

Using our formula, p hat equals 0.18 or 18%, the point estimate for the proportion, q hat equals 1 minus p hat, so 1 minus 0.18, which is 0.82, E, the margin of error is 3% or 0.03, how accurate we want to be, and z alpha over 2 for 90% is 1.645.

Now we can do some fun mathematical calculations and we get 443.79, rounding up to 444 as the sample size needed.

Now, lets say same question, but we are going to assume there was no previous study done, so what is the minimum sample size needed?

Again using our formula, the values for p hat and q hat are going to change to 0.5 for each. Again, fun calculations gives us 752 as the sample size.

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