Question

A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions. (Round your answers up to the nearest integer.)

(a) What sample size is necessary if the 95% CI for p is to have a width of at most 0.12 irrespective of p?

(b) If the legislator has strong reason to believe that at least 3/4 of the electorate know of her position, how large a sample size would you recommend to maintain a width of at most 0.12?

Answer 1

Solution:

a) Given ,

c = 95% = 0.95

Width of the interval is 0.12

Margin of error E is half of the width of the interval

So , E = 0.12/2 = 0.06

Here , no prior information of p is available. In this case we take

p = 0.5 and 1 - p = 1 - 0.5 = 0.5

Now,

= 1 - c = 1 - 0.95 = 0.05

/2 = 0.025

   = 1.96 .. (using z table)

The sample size for estimating the proportion is given by

n =

= (1.96)² * 0.5 * 0.5 / (0.06²)

= 266.777777778

= 267 ....(round to the next whole number)

Answer : Necessary sample size is 267

b) Here , p = 3/4 = 0.75

So , 1 - p = 1 - 0.75 = 0.25

The sample size for estimating the proportion is given by

n =

= (1.96)² * 0.75 * 0.25 / (0.06²)

=  200.083333333

= 201    ....(round to the next whole number)

Answer : Necessary sample size is 201