Question

A simple random sample of size n=300 individuals who are currently employed is asked if they work at home at least once per week. Of the 300 employed individuals​ surveyed, 38 responded that they did work at home at least once per week. Construct a​ 99% confidence interval for the population proportion of employed individuals who work at home at least once per week.

The lower bound is

The upper bound is

​(Round to three decimal places as​ needed.)

Answer 1

Solution:

Given:

n = Number of  individuals who are currently employed is asked if they work at home at least once per week

n = 300

x = Number of employed individuals​ surveyed responded that they did work at home at least once per week.

x = 38

We have to construct a​ 99% confidence interval for the population proportion of employed individuals who work at home at least once per week.

Formula:

where

and

Z_c is z critical value for c = 0.99 confidence level.

Find Area = ( 1+c)/2 = ( 1 + 0.99 ) / 2 = 1.99 /2 = 0.9950

Thus look in z table for Area = 0.9950 or its closest area and find corresponding z critical value.

From above table we can see area 0.9950 is in between 0.9949 and 0.9951 and both are at same distance from 0.9950, Hence corresponding z values are 2.57 and 2.58

Thus average of both z values is 2.575

Thus Z_c = 2.575

thus

thus

The lower bound is : 0.077

The upper bound is : 0.176