Question

In a certain region, 24% of people over age 50 didn't graduate from high school. We would like to know if this percentage is the same among the 25-30 year age group. Use critical values to exactly 3 decimal places.

(a) How many 25-30 year old people should be surveyed in order to estimate the proportion of non-grads to within 6% with 99% confidence?

(b) Suppose we wanted to cut the margin of error to 2%. How many people should be sampled now?  

(c) What sample size is required for a margin of error of 3%?  

Answer 1

Solution :

Given that,

= 24% = 0.24

1 - = 1 - 0.24 = 0.76

At 99% confidence level the z is ,

  = 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z_/2 = Z_0.005 = 2.576

(a)

Margin of error = E = 6% = 0.06

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (2.576 / 0.06)² * 0.24 * 0.76

= 336.21 = 337

337 old people 25-30 year should be surveyed .

(b)

Margin of error = E = 2% = 0.02

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (2.576 / 0.02)² * 0.24 * 0.76

= 3025.91 = 3026

3026 people should be sampled .

(c)

Margin of error = E = 3% = 0.03

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (2.576 / 0.03)² * 0.24 * 0.76

= 1344.85 = 1345

Sample size = 1345