Let p denote the proportion of elderly people with artificial hips. In an earlier study, the estimated proportion of the number of elderly people with artificial hips was found to be 11%.

Gretchen Allison

Gretchen Allison

Answered question

2022-09-10

This is a question from a sample mid-term for a first course in Statistics
Let p denote the proportion of elderly people with artificial hips. In an earlier study, the estimated proportion of the number of elderly people with artificial hips was found to be 11%.
Q.1. Using earlier study's estimate, find the number of samples we would need in order to create a 95 confidence interval for the proportion of elderly people with artificial which is no more than 0.05 away from the true proportion.
Ans. I have no clue how to do this. If could get the formula on how to approach this I would appreciate it.
Q.2. What is the maximum sample size needed to create a 93 confidence interval for the proportion of elderly people with artificial hips which is no more than 0.04 away from the true proportion ? Ans I would appreciate a formula for this as well.

Answer & Explanation

ignaciopastorp6

ignaciopastorp6

Beginner2022-09-11Added 14 answers

Step 1
The formula for the minimum required sample size is
n = z 2 p ( 1 p ) d 2 ,
where z = z α / 2 and p is the true proportion and d is half the length of the ( 1 α ) % confidence interval.
So in exercises like these you
1. determine α, e.g. if you're looking at 95% confidence intervals, then set α = 0.05. This determines z,
2. find d which is half the length of the confidence interval you want to obtain.
3. and then use the formula above with p being replaced by some estimate (usually obtained by earlier studies or smaller pilot studies).
Often you're not explicitly given d as half the length of the confidence interval, but rather as "create a 95% confidence interval such that the estimated proportion is no more than d away from the true proportion".

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