How large of a sample size is required?

Question

You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately p∗=78%. You would like to be 95% confident that your estimate is within 0.1% of the true population proportion. How large of a sample size is required?

Do not round mid-calculation. However, use a critical value accurate to three decimal places.

Answer 1

Hi Kaitlyn,

This is like the one we did before; only instead of finding the margin of error, we're looking for he sample size n.

Recall that the formula for margin of error is:

MoE= z*sqrt(p(1-p)/n)

Breaking this down:

z*=z-critical value

p=probability you were given

n=sample size we are looking for

Now, for this problem:

MoE=0.001 (decimal equivalent to 0.1%)

z*=1.96, z-critical value for 95% confidence, worth memorizing if possible

p=0.78

n=n

Substituting:

0.001=1.96*sqrt((0.78*0.22)/n))

Divide both sides by 1.96:

0.00051=sqrt((0.78*0.22)/n))

Square both sides:

0.00051²=(0.78*0.22)/n

Multiply both sides by n:

(0.00051²*n)=(0.78*0.22)

0.00051²n=0.1716

n=0.1716/0.00051²

n=659747

This is extremely high because your margin of error--0.1%--is extremely low. I hope this helps.