Question

Explain in your own words why the margin of error is the largest when p is 0.5, and how many participants would we need to ensure that the margin of error is smaller than 0.05 using a 95% confidence interval.

Answer 1

Solution:

Formula for sample size n for proportion is:

In this formula we do multiplication of p and ( 1-p)

This product is maximum when p = 0.5 which is = 0.5 X ( 1-0.5) = 0.5 X 0.5 = 0.25 and for any other value of p, multiplication of p and (1-p) is less than 0.25

Lets consider p = 0.45 , then 0.45 X ( 1 - 0.45) = 0.45 X 0.55 = 0.2475 < 0.25

Lets consider p = 0.49 , then 0.49 X ( 1 - 0.49) = 0.49 X 0.51 = 0.2499 < 0.25

Lets consider p = 0.98 , then 0.98 X ( 1 - 0.98) = 0.98 X 0.02 = 0.0196< 0.25

thus for any other value of p, product is < 0.25

Hence sample size is largest when p = 0.5

Now find sample size n for:

E = Margin of Error = 0.05

c = confidence level = 95%

Since p is unknown, we use p = 0.5

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

( Sample size is always rounded up)