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 28 ​% 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

Step 1: Using the data given in the question, figure out the following variables:

za/2: Divide the confidence interval by two, and look that area up in the z-table:
.99 / 2 = 0.495
The closest z-score for 0.475 is 2.58
E (margin of error): Divide the given width by 2.
2% / 2
= 0.02 / 2
= 0.01
[phat] : If you aren’t given phat, use 50%.
[qhat] : subtract [phat] from 1.
1 – 0.50 = 0.50

Step 2:Multiply [phat] by [qhat] . Set this number aside for a moment.
0.5 × 0.5 = 0.25

Step 3: Divide Za/2 by E.
2.58 / .01 = 258

Step 4: Square Step 3:
258 × 258= 66564

Step 5: Multiply Step 2 by Step 4:
0.25 × 66564= 16641
= 16641 minimum sample size.

b) phat= 0.28 and qhat= 1-0.28= 0.72

Multiply [phat] by [qhat] . Set this number aside for a moment.
0.28 × 0.72 = 0.2016

0.2016*66564= 13419.3024

13419 minimm sample size.

c) When we asssmed phat=0.5 we get sample size 16641

when we found that 0.28 p hat we get sample size 13419

as phat reduced sample size also got reduced.

NOTE: If answer help you to understand then please thumbs up. Thank you.