Question

Human Resource Consulting (HRC) surveyed a random sample of 53 Twin Cities construction companies to find information on the costs of their health care plans. One of the items being tracked is the annual deductible that employees must pay. The Minnesota Department of Labor reports that historically the mean deductible amount per employee is $501 with a standard deviation of $102.

1. Point vs. Interval Estimation

Here you will discuss the importance of constructing confidence intervals for the population mean. You want to be sure to address these questions:

• What are confidence intervals?
• What is a point estimate?
• What is the best point estimate for the population mean? Explain why it is the best.
• Why do we need confidence intervals?

2. Mean Estimation

First clearly give the best point estimate for the population mean the data was drawn from. Next you will construct two confidence intervals: a 95% confidence interval and a 99% confidence interval. Bear in mind that the population standard deviation is unknown, but you may assume that the population is approximately normally distributed. Please show your work for the construction of these confidence intervals. If you used technology or a calculator please explain what you used and how you got your intervals.

3. Interpretation and Conclusion

Compare and contrast your findings for the 95% and 99% confidence intervals. Did you notice any changes in your interval estimate? Be specific. What conclusion(s) can be drawn about your interval estimates when the confidence level is increased? Explain why you draw this conclusion.

Answer 1

1. Point vs. Interval Estimation

What are confidence intervals?

A confidence interval is a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unknown population parameter.

What is a point estimate?

It is a single value given as an estimate of a parameter of a population.

What is the best point estimate for the population mean?

Sample mean is the best point estimate for the population mean because it has the lowest variance of all the estimates and also it is an unbiased estimate of population mean.

Why do we need confidence intervals?

The purpose of confidence intervals is to give us a range of values for our estimated population parameter rather than a single value or a point estimate. The estimated confidence interval gives us a range of values within which we believe with certain probability (confidence level), that the true population value falls.

2. Mean Estimation

95% confidence interval :

99% confidence interval :

3. Interpretation and Conclusion

We can see that 95% confidence interval is narrower than 99% confidence interval.

Hence, as the confidence level increases the width of the confidence interval also increases.

So, If we require a narrow confidence interval we need to take a lower confidence interval as well.