You would like to be 99% confident that you estimate is within 4% of the true population proportion. How large of a sample size is required?

Question

You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable preliminary estimation for the population proportion.

Answer 1

To solve this problem, we use the formula for determining sample size for estimating a population proportion. The formula is:

n = (Z² * p * (1 - p)) / E²

where:

1. n is the sample size.
2. Z is the Z-score corresponding to the desired confidence level.
3. p is the estimated population proportion.
4. E is the margin of error.

In this scenario:

1. You want a 99% confidence level. The Z-score for 99% confidence is approximately 2.576 (this value comes from a standard Z-score table).
2. Since you have no preliminary estimate for the population proportion, a conservative approach is to use p = 0.5. This is because the product p * (1 - p) reaches its maximum at p = 0.5, which gives the largest sample size ensuring coverage for any proportion.
3. The desired margin of error (E) is 4%, or 0.04 in decimal form.

Plugging these values into the formula gives:

n = (2.576² * 0.5 * (1 - 0.5)) / 0.04²

Upon calculation, the required sample size (n) is approximately 1037. This means you would need a sample of at least 1037 individuals to be 99% confident that your estimate is within 4% of the true population proportion, assuming no prior knowledge of the population proportion. This size ensures the most conservative estimate, covering all possible scenarios of the true population proportion.

Answer 2

Hi Lauren,

The formula for this is:

n= (z*/m)²*p*(1-p*)

Breaking this down:

n=desired sample size

z*=z-critical value

m=margin of error

p*=estimate of population proportion

Now, for our problem:

z*=2.576 (Critical value likely worth memorizing. Also note z*₉₅=1.96, z*₉₀=1.645)

m=0.04

p*=0.5–When you do not know anything about a population in advance, always go with 0.5 as the estimate.

Thus:

n=(2.576/0.04)²(0.5*0.5)

n=1036.84, round up:

n=1037

I hope this helps.