Question

In a sample of 250 people, only 45 are observed washing their hands after using the bathroom. Con-
struct a 99% confidence interval for p, the proportion of people who wash their hands after using the bathroom.

Answer 1

Solution:
Given in the question
Number of sample. (n)= 250 people
Number of Favourable cases (X)= 45
Sample proportion P^= X/n = 45/250 = 0.18
Popint estimate = 0.18
We need to calculate 99% confidence interval for p, who wash their hands after using the bathroom, which can be calculated as
Here we will use one sample propotion Z test to calculate confidence interval:
Point estimate +/- Zalpha/2 *Sqrt(p^(1-p^)/n)
Here Confidence level = 0.99
Level of significance(alpha) = 1 - Confidence level = 1 - 0.99 = 0.01
Alpha/2 = 0.005
From Z table we found Zalpha/2 = 2.5758
So 99% confidence interval is
0.18 +/- 2.5758*sqrt(0.18*(1-0.18)/250)
0.18 +/- 2.5758*0.0243
0.18 +/- 0.063
0.117 to 0.243
So we are 99% confident that proportion of people who wash their hands after using the bathroom is 0.117 to 0.243