It is said that healthy and happy workers are efficient and productive. A

company that manufactures exercising machines wanted to know the
percentage of large companies that provided on-site health club facilities.
A sample of 240 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 such facilities on site?

B. Construct a 97% confidence interval for the percentages of all such
companies that provide such facilities on site.

To find the point estimate and construct a confidence interval, we need to use statistical calculations. Here's how you can do it:

A. Point Estimate:
To find the point estimate of the percentage of all such companies that provide on-site health club facilities, we divide the number of companies found in the sample that provide such facilities (96) by the total number of companies in the sample (240). The formula for calculating the point estimate is:

Point Estimate = (Number of companies providing facilities / Total number of companies) * 100

So, the point estimate is:

Point Estimate = (96 / 240) * 100 = 40%

Therefore, the point estimate is 40%.

B. Confidence Interval:
To construct a confidence interval, we can use the formula:

Confidence Interval = Point Estimate ± Margin of Error

The margin of error depends on the confidence level and the sample size. In this case, we want a 97% confidence interval, so we need to find the critical value associated with a 97% confidence level. We can use a z-table or calculator to find the critical value. The critical value for a 97% confidence level is approximately 2.17.

The margin of error is calculated using the following formula:

Margin of Error = Critical value * Standard Error

The standard error can be calculated as:

Standard Error = sqrt((Point Estimate * (1 - Point Estimate)) / Sample Size)

Plugging in the values from our problem:

Sample Size = Total number of companies = 240
Point Estimate = 40%
Critical value = 2.17

Now, let's calculate the margin of error:

Standard Error = sqrt((0.4 * (1 - 0.4)) / 240) ≈ 0.0346

Margin of Error = 2.17 * 0.0346 ≈ 0.0751

So, the margin of error is approximately 0.0751.

Finally, we can construct the confidence interval:

Confidence Interval = Point Estimate ± Margin of Error

Confidence Interval = 40% ± 0.0751

Confidence Interval = (39.9249%, 40.0751%)

Therefore, the 97% confidence interval for the percentage of all such companies that provide on-site health club facilities is approximately (39.9249%, 40.0751%).

A. The point estimate of the percentage of all companies that provide on-site health club facilities can be calculated by dividing the number of companies in the sample that provide such facilities (96) by the total number of companies in the sample (240), and then multiplying the result by 100 to get a percentage:

Point estimate = (96 / 240) * 100 = 40%.

Therefore, the point estimate is 40%.

B. To construct a 97% confidence interval for the percentage of all companies that provide on-site health club facilities, we can use the formula:

Confidence interval = Point estimate ± (Critical value * Standard error)

The critical value is determined using the desired confidence level (97%) and the sample size (240).

To find the critical value, we can refer to a standard normal distribution table or use statistical software. For a 97% confidence level, the critical value is approximately 2.17.

The standard error can be calculated using the following formula:

Standard error = sqrt((Point estimate * (1 - Point estimate)) / Sample size)

Standard error = sqrt((0.40 * (1 - 0.40)) / 240) ≈ 0.0316

Now we can substitute the values into the formula to calculate the confidence interval:

Confidence interval = 0.40 ± (2.17 * 0.0316)

Confidence interval = 0.40 ± 0.0686

Confidence interval ≈ [0.3314, 0.4686]

Therefore, the 97% confidence interval for the percentage of all companies that provide on-site health club facilities is approximately 33.14% to 46.86%.

A) Point of estimate = phat = 96/240 = 0.4

za/2 = (1-.97)/2 = 0.015
z0.015 = 2.17
Sdv = sqrt(p*q/n) = 0.0316

0.4 -+ 2.17* 0.0316

[0.3314, 0.4686] or 33.14% to 46.86%