Question

A researcher wishes to​ estimate, with 90​% ​confidence, the population proportion of adults who are confident with their​ country's banking system. His estimate must be accurate within 4​% of the population proportion. ​(a) No preliminary estimate is available. Find the minimum sample size needed. ​(b) Find the minimum sample size​ needed, using a prior study that found that 32​% of the respondents said they are confident with their​ country's banking system. ​(c) Compare the results from parts ​(a) and ​(b).

Answer 1

Solution :

Given that,

a)

= 1 - = 0.5

margin of error = E = 4% = 0.04

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

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

= (1.645 / 0.04)² * 0.5 * 0.5

= 422.8 = 423

sample size = 423

b)

= 32% = 0.32

1 - = 1 - 0.32 = 0.68

margin of error = E = 0.04

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

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

= (1.645 / 0.04)² * 0.32 * 0.68

= 368

sample size = 368

c)

Having an estimate of the population proportion reduces the minimum sample size is needed .