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 240 such companies showed that 96 of them provide such facilities on site. Construct a 98% confidence interval for the percentage of all such companies that provide such facilities on site. Your answer

Question

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 240 such companies showed that 96 of them provide such facilities on site. Construct a 98% confidence interval for the percentage of all such companies that provide such facilities on site. Your answer

Answer 1

To construct a confidence interval for the percentage of all large companies that provide on-site health club facilities, we can use the formula for calculating a confidence interval for a proportion. The formula is:

Confidence Interval = sample proportion ± (critical value * standard error)

First, let's calculate the sample proportion:
sample proportion = number of companies providing facilities / total number of companies in the sample = 96/240 = 0.4

Next, we need to determine the critical value associated with a 98% confidence level. For a confidence level of 98%, the alpha value (α) is 1 - 0.98 = 0.02. Since this distribution is approximately normal, we can find the critical value using a z-table or a calculator. The critical value for a two-tailed test with α = 0.02 is approximately 2.33.

Now, we need to calculate the standard error. The formula for standard error of a proportion is sqrt((p * (1-p)) / n), where p is the sample proportion and n is the sample size.

standard error = sqrt((0.4 * (1-0.4)) / 240) ≈ 0.032

Finally, we can calculate the confidence interval:

Confidence Interval = 0.4 ± (2.33 * 0.032)

Lower bound = 0.4 - (2.33 * 0.032) ≈ 0.328
Upper bound = 0.4 + (2.33 * 0.032) ≈ 0.472

So, the 98% confidence interval for the percentage of all large companies that provide on-site health club facilities is approximately 32.8% to 47.2%. This means that we can be 98% confident that the true proportion of all large companies providing such facilities falls within this range.