Question

Determine the sample size n needed to construct a 95​% confidence interval to estimate the population proportion for the following sample proportions when the margin of error equals 4​%.

a. p over bar equals 0.10

b. p over bar equals 0.20

c. p over bar equals 0.30

Answer 1

Solution :

Given that,

= 0.10

1 - = 1 - 0.10 = 0.9

margin of error = E = 4% = 0.04

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96 ( Using z table )

Sample size = n = (Z/2 / E)2 * * (1 - )

= (1.96 / 0.04)2 * 0.10 * 0.9

= 216.09

Sample size =216

b.

Solution :

Given that,

= 0.20

1 - = 1 - 0.20 = 0.8

margin of error = E = 4% = 0.04

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96 ( Using z table )

Sample size = n = (Z/2 / E)2 * * (1 - )

= (1.96 / 0.04)2 * 0.2 * 0.8

= 385

Sample size =382

c.

Solution :

Given that,

= 0.30

1 - = 1 - 0.30 = 0.7

margin of error = E = 4% = 0.04

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96 ( Using z table )

Sample size = n = (Z/2 / E)2 * * (1 - )

= (1.96 / 0.04)2 * 0.30* 0.70

= 504.21

Sample size =505