91Ó°ÊÓ

Briefly explain how the width of a confidence interval decreases with a decrease in the confidence level. Give an example.

For a data set obtained from a sample, \(n=20\) and \(\bar{x}=24.5\). It is known that \(\sigma=3.1\). The population is normally distributed. a. What is the point estimate of \(\mu\) ? b. Make a \(99 \%\) confidence interval for \(\mu\). c. What is the margin of error of estimate for part b?

For a population, the value of the standard deviation is 4.96. A sample of 32 observations taken from this population produced the following data. \(\begin{array}{llllllll}74 & 85 & 72 & 73 & 86 & 81 & 77 & 60 \\ 83 & 78 & 79 & 88 & 76 & 73 & 84 & 78 \\ 81 & 72 & 82 & 81 & 79 & 83 & 88 & 86 \\ 78 & 83 & 87 & 82 & 80 & 84 & 76 & 74\end{array}\) a. What is the point estimate of \(\mu\) ? b. Make a \(99 \%\) confidence interval for \(\mu .\) c. What is the margin of error of estimate for part b?

A bank manager wants to know the mean amount of mortgage paid per month by homeowners in an area. A random sample of 120 homeowners selected from this area showed that they pay an average of \(\$ 1575\) per month for their mortgages. The population standard deviation of such mortgages is \(\$ 215\). a. Find a \(97 \%\) confidence interval for the mean amount of mortgage paid per month by all homeowners in this area. b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which alternative is the best?

You are interested in estimating the mean commuting time from home to school for all commuter students at your school. Briefly explain the procedure you will follow to conduct this study. Collect the required data from a sample of 30 or more such students and then estimate the population mean at a \(99 \%\) confidence level. Assume that the population standard deviation for such times is \(5.5\) minutes.

What assumptions must hold true to use the \(t\) distribution to make a confidence interval for \(\mu\) ?

a. A sample of 400 observations taken from a population produced a sample mean equal to \(92.45\) and a standard deviation equal to \(12.20 .\) Make a \(98 \%\) confidence interval for \(\mu\). b. Another sample of 400 observations taken from the same population produced a sample mean equal to \(91.75\) and a standard deviation equal to \(14.50 .\) Make a \(98 \%\) confidence interval for \(\mu .\) c. A third sample of 400 observations taken from the same population produced a sample mean equal to \(89.63\) and a standard deviation equal to \(13.40 .\) Make a \(98 \%\) confidence interval for \(\mu\). d. The true population mean for this population is \(90.65 .\) Which of the confidence intervals constructed in parts a through c cover this population mean and which do not?

A random sample of 16 airline passengers at the Bay City airport showed that the mean time spent waiting in line to check in at the ticket counters was 31 minutes with a standard deviation of 7 minutes. Construct a \(99 \%\) confidence interval for the mean time spent waiting in line by all passengers at this airport. Assume that such waiting times for all passengers are normally distributed.

Check if the sample size is large enough to use the normal distribution to make a confidence interval for \(p\) for each of the following cases. a. \(n=50\) and \(\hat{p}=.25\) b. \(n=160\) and \(\hat{p}=.03\) c. \(n=400\) and \(\hat{p}=.65\) d. \(n=75 \quad\) and \(\quad \hat{p}=.06\)

A drug that provides relief from headaches was tried on 18 randomly selected patients. The experiment showed that the mean time to get relief from headaches for these patients after taking this drug was 24 minutes with a standard deviation of \(4.5\) minutes. Assuming that the time taken to get relief from a headache after taking this drug is (approximately) normally distributed, determine a \(95 \%\) confidence interval for the mean relief time for this drug for all patients.