Posted by Steven on .
A company that manufactures exercise machines wanted to know the percentage of large companies that provides on-site health club facilities. A sample of 240 such companies showed that 96 of them provide such facilities on site.
a. What is the point estimate of the percentage of all such companies that provide these facilities? Explain briefly the concept of the point estimate.
b. Construct a 97% confidence interval for the percentage of all such companies.
c. What is the margin of error for this estimate?
d. If the company wanted a narrower interval, name two things it could do.
e. Which is the better strategy and why?
f. What does it mean to have a 97% confidence in the interval?
The point estimate is an estimate of the population percentage given the sample you are working with.
Percentage = 96/240 = p-hat
q-hat = 1 - p-hat
97% confidence interval means that you have 3% left which is split between two tails. You need to split .03 in half to get .015. Find that value in the body of the z-table and use that value to get the z-score from the margins of the table.
You then calculate the margin of error using, p-hat, q-hat and n along with z. That answer is the number that you add or subtract to p- hat value to get the confidence interval.
To get a narrow interval, you can adjust the confidence interval.. ex. 95% gives you a narrower interval. You can also adjust the sample size.
People often confuse the meaning of a 97% confidence interval. It means that if you took many, many samples from the same population and found the percentage, calculated the confidence intervals in each case then 97% of those confidence intervals that you constructed will contain the true population percentage.
You cannot say that there is a 97% chance that this value will be within the 97% or that it represents the true population mean.