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?
p = 96/240 q = (240-96)/240
n =240
a. find p
b. Find the z-value for a 97% confidence interval. You will most likely have to find the tails which is 3%/2 or 1.5% or .015
c. The margin of error is the +/- value that you will add to the p that you found originally.
d. For a narrower interval, you can use a smaller confidence and/ or do something with the n. (Do you know what that is)
Most people misinterpret what 97% confidence really means...
If you had the same population and took the sample of the same size over and over and over again. Did the math and created many, many, many confidence intervals then 97% of those intervals that you created will contain the true population mean.
97% does not mean that you have a 97% chance that the p that you found is correct.
a. The point estimate of the percentage of all such companies that provide on-site health club facilities can be calculated by dividing the number of companies in the sample that provide these facilities by the total number of companies in the sample. In this case, the point estimate would be 96/240, which is 0.4 or 40%.
The concept of a point estimate is to use a single value to estimate an unknown population parameter. It provides an estimate based on the available data, but it is important to note that it may not be exactly equal to the true population value.
b. To construct a confidence interval, we need to determine the standard error of the proportion and use it to calculate the interval estimate. In this case, the formula to calculate the confidence interval for a proportion is given by:
p̂ ± z * √((p̂(1-p̂))/n)
where p̂ is the point estimate, z is the z-score corresponding to the desired confidence level (97% in this case), and n is the sample size.
c. The margin of error for this estimate is the distance between the point estimate and the upper or lower bound of the confidence interval. It indicates the maximum likely difference between the point estimate and the true population parameter.
d. If the company wanted a narrower interval, it could either increase the sample size or choose a smaller confidence level. Increasing the sample size will reduce the standard error and result in a narrower interval, while choosing a smaller confidence level will decrease the critical value (z-score) and also narrow the interval.
e. The better strategy depends on the specific needs and requirements of the company. If accuracy is more important and the cost or effort involved in obtaining a larger sample is not a concern, increasing the sample size would be a good option. However, if the company wants a narrower interval while maintaining a higher level of confidence, choosing a smaller confidence level would be more suitable.
f. Having a 97% confidence in the interval means that if we were to repeat the sampling process multiple times, 97% of the resulting confidence intervals would contain the true population parameter. In other words, we can say with 97% confidence that the true value of the percentage of all such companies that provide on-site health club facilities lies within the calculated interval. The higher the confidence level, the wider the interval will be, providing a greater level of certainty.