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?

n equals ​(Round up to the nearest whole number as​ needed.)

Answer 1

Solution: 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.

Answer: The minimum sample size needed when there no preliminary estimate of population proportion is:

Where

is the z critical value at 0.05 significance level and is given below:

is the margin of error and is % or

Therefore the minimum sample size needed is

​(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.

Answer: The minimum sample size needed is:

Where

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

Therefore, we have:

  

Therefore the minimum sample size, using a prior study estimate of population proportion is

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

Answer: We clearly see the sample size needed is less when we had prior estimate of the proportion given () compared to sample size needed when we had no prior information. Therefore we can state as the estimate of the proportion decreases the sample size needed will also decrease.