Question

A researcher wishes to​ estimate, with 95​% ​confidence, the population proportion of adults who are confident with their​ country's banking system. His estimate must be accurate within 5​% of the population proportion.

​(a) No preliminary estimate is available. Find the minimum sample size needed.

n=

​(b) Find the minimum sample size​ needed, using a prior study that found that 22​%

of the respondents said they are confident with their​ country's banking system.

n=

​(c) Compare the results from parts ​(a) and choose one

A.Having an estimate of the population proportion has no effect on the minimum sample size needed.

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

C.Having an estimate of the population proportion raises the minimum sample size needed.

Answer 1

Solution :

Given that,

margin of error = E = 5% = 0.05

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.96

A)

= 0.5

1 - = 1 - 0.5 = 0.5

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

= (1.96 / 0.05)² * 0.5 * 0.5

= 384.13 = 384

sample size = 384

B)

= 22% = 0.22

1 - = 1 - 0.22 = 0.78

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

= (1.96 / 0.05)² * 0.22 * 0.78

= 263.68 = 264

sample size = 264

C)

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