It is said that happy and healthy workers are efficient and productive. A company that manufactures exercising machines wanted to know the percentage of large companies that provide on-site health club facilities. A sample of 200 such companies showed that 150 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 98% confidence interval for the percentage of all such companies that provide such facilities on site.
C. What is the margin of error for this estimate?
To answer these questions, we can use statistical techniques, specifically point estimation and confidence intervals.
A. Point Estimate:
The point estimate is an estimate of a population parameter based on the sample data. In this case, the population parameter is the percentage of all large companies that provide on-site health club facilities. The point estimate is calculated by dividing the number of companies that provide on-site health club facilities by the total number of companies in the sample.
So, the point estimate of the percentage of all such companies that provide on-site health club facilities is:
(150/200) * 100 = 75%
Therefore, the point estimate is 75%.
B. Confidence Interval:
The confidence interval is used to estimate the range within which the true population parameter lies with a certain level of confidence. In this case, we want to construct a 98% confidence interval for the percentage of all companies that provide on-site health club facilities.
To construct the confidence interval, we can use the following formula:
Confidence Interval = Point Estimate ± Margin of Error
The margin of error can be calculated using the formula:
Margin of Error = Z * (sqrt(p(1 - p)) / n)
Where:
- Z is the z-score corresponding to the desired confidence level (98% in this case)
- p is the point estimate (75% in this case)
- n is the sample size (200 in this case)
First, we need to find the value of Z. For a 98% confidence level, the corresponding Z-score can be obtained from the standard normal distribution table, which gives us a value of approximately 2.33.
Next, we plug in the values into the margin of error formula:
Margin of Error = 2.33 * (sqrt(0.75 * 0.25) / 200)
Calculating this, we get:
Margin of Error ≈ 0.0676
Finally, we can construct the confidence interval:
Confidence Interval = 75% ± 0.0676
The lower bound of the confidence interval is 75% - 0.0676 = 74.9324, and the upper bound is 75% + 0.0676 = 75.0676.
Therefore, the 98% confidence interval for the percentage of all such companies that provide on-site health club facilities is approximately (74.9324%, 75.0676%).
C. Margin of Error:
The margin of error represents the range around the point estimate within which the true population parameter is likely to fall. In this case, we calculated the margin of error as 0.0676. This means that we can be 98% confident that the true percentage of all companies providing on-site health club facilities is within 0.0676 of the point estimate.
So, the margin of error for this estimate is approximately 0.0676.