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.

a. The point estimate is the estimate of a population parameter based on a sample statistic. In this case, the point estimate for the percentage of all companies providing on-site health club facilities is 40% (96 out of 240).

b. To construct a confidence interval, we can use the formula:

Confidence Interval = Point Estimate ± Margin of Error

Since the sample size is large (240), we can use the normal approximation to the binomial distribution. The formula for the margin of error is:

Margin of Error = Z * sqrt((p̂ * q̂) / n)

Here, p̂ is the sample proportion (96/240), q̂ is the complement of p̂ (1 - p̂), n is the sample size (240), and Z represents the critical value for a given level of confidence (in this case, 97%). The critical value can be obtained from the standard normal distribution table or using statistical software.

c. The margin of error is the range around the point estimate within which the true population parameter is likely to fall. It is calculated using the formula mentioned in part b.

d. To obtain a narrower interval, the company could either increase the sample size or choose a lower level of confidence. By increasing the sample size, more data would be collected and the estimate would become more precise. However, this might require additional resources. Choosing a lower level of confidence (e.g., 90% instead of 97%) would result in a narrower interval, but it would also reduce the confidence level and introduce a higher risk of being incorrect.

e. The better strategy depends on the company's specific needs and resources. If the company wants a higher degree of confidence in the interval and has the resources to collect a larger sample, increasing the sample size would be the better strategy. On the other hand, if the company wants a narrower interval and is willing to accept a slightly higher risk of being incorrect, choosing a lower level of confidence would be the better strategy.

f. A 97% confidence interval means that if we repeated the sampling process multiple times and constructed intervals in the same way, around 97% of these intervals would contain the true population parameter (percentage of large companies providing on-site health club facilities). In other words, there is a high level of certainty that the true value falls within the calculated interval. However, there is still a small chance (3%) that the interval does not capture the true value.