Question

A researcher wishes to​ estimate, with 99​% ​confidence, the population proportion of adults who think Congress is doing a good or excellent job. Her estimate must be accurate within 2​% of the true 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 34​% of the respondents said they think Congress is doing a good or excellent job. ​(c) Compare the results from parts​ (a) and​ (b).

Answer 1

E = Margin of error = 2% = 0.02

Confidence level = 99% = 0.99

Using Z table critical value for 99% is

Z = 2.58      ( a = 1-0.99= 0.01 , area = 1-(a/2) = 0.995 check 0.995 is Z or table or use excel command

                             =NORMSINV(0.995)    )

Part a.

no preliminary estimate is available so we assume,

p = 0.5

Sample size formula :

n = 0.25 * 16641

n = 4160.25

n = 4160       ( rounded value )

Part b.

A prior study that found that 34​% of the respondents said they think Congress is doing a good or excellent job.

p = 0.34

Sample size formula :

n = 0.2244 * 16641

n = 3734.24

n = 3734       ( rounded value )

Partc.

When prior estimate is not available we get heigher sample size is 4160 and for prior size 34% the sample size is going to decrease it is 3734.

At prior estimate 50% we get heigher sample size.