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 2​% 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 22​% of the respondents said they are confident with their​ country's banking system. ​

(c) Compare the results from parts ​(a) and ​(b).

​(a) What is the minimum sample size needed assuming that no prior information is​ available? nequals nothing ​(Round up to the nearest whole number as​ needed.)

​(​b) What is 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? nequals nothing ​(Round up to the nearest whole number as​ needed.)

Answer 1

Here:-

Margin of error (E) = 2%

E = 0. 02

Confidence is 95%

a = 0.05

z = 1.96

a. No preliminary estimate is available so assume p is 0.5

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

= 0.5 * 0.5 * ( 1.96 / 0.02 )²

= 0.5 * 0.5 * 9604

= 2401

Thus, sample size needed is 2401 adults.

b. Here, p is already given 22%

p = 0.22

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

= 0.22 * 0.78 * ( 1.96 / 0.02 )²

= 0.22 * 0.78 * 9604

= 1648.046

Thus, minimum sample size needed is 1648 adults, when a prior study is used that 22​% of the respondents said they are confident with their​ country's banking system. ​

c. It is clear that, the value of the minimum sample size when there is no preliminary estimate available and considering the population proportion is 2401. And the value of the sample size having 22% of the respondents said they are confident with their countries banking system is found to be 1648

Therefore, it is clear that the sample size obtained by using the proportion value from the prior study is small compared to the sample size obtained by using no preliminary estimate. There is effect of estimated population proportion on the minimum sample size needed. The increase in population proportion increases the sample size.

Thus, having an estimate of the population proportion reduces the minimum sample size needed. The sample size found using the no preliminary estimate is larger than sample obtained by using the prior study.