91Ó°ÊÓ

If the original \(x\) distribution has a relatively small standard deviation, the confidence interval for \(\mu\) will be relatively short.

How much does a sleeping bag cost? Let's say you want a sleeping bag that should keep you warm in temperatures from \(20^{\circ} \mathrm{F}\) to \(45^{\circ} \mathrm{F}\). A random sample of prices (\$) for sleeping bags in this temperature range was taken from Backpacker Magazine: Gear Guide (Vol. 25 , Issue 157, No. 2). Brand names include American Camper, Cabela's, Camp 7, Caribou, Cascade, and Coleman. \(\begin{array}{rrrrrrrrrr}80 & 90 & 100 & 120 & 75 & 37 & 30 & 23 & 100 & 110 \\ 105 & 95 & 105 & 60 & 110 & 120 & 95 & 90 & 60 & 70\end{array}\) (a) Use a calculator with mean and sample standard deviation keys to verify that \(\bar{x} \approx \$ 83.75\) and \(s \approx \$ 28.97\). (b) Using the given data as representative of the population of prices of all summer sleeping bags, find a \(90 \%\) confidence interval for the mean price \(\mu\) of all summer sleeping bags.

How much do wild mountain lions weigh? The 77 th Annual Report of the New Mexico Department of Game and Fish, edited by Bill Montoya, gave the following information. Adult wild mountain lions (18 months or older) captured and released for the first time in the San Andres Mountains gave the following weights (pounds): \(\begin{array}{llllll}68 & 104 & 128 & 122 & 60 & 64\end{array}\) (a) Use a calculator with mean and sample standard deviation keys to verify that \(\bar{x}=91.0\) pounds and \(s \approx 30.7\) pounds. (b) Find a \(75 \%\) confidence interval for the population average weight \(\mu\) of all adult mountain lions in the specified region.

Do you want to own your own candy store? Wow! With some interest in running your own business and a decent credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Express, The Fudge Company, Karmel Corn, and Rocky Mountain Chocolate Factory. Startup costs (in thousands of dollars) for a random sample of candy stores are given below (Source: Entrepreneur Magazine, Vol. 23, No. 10\()\). \(\begin{array}{lllllllll}95 & 173 & 129 & 95 & 75 & 94 & 116 & 100 & 85\end{array}\) Use a calculator with mean and sample standard deviation keys to verify that \(\bar{x} \approx 106.9\) thousand dollars and \(s \approx 29.4\) thousand dollars. Find a \(90 \%\) confidence interval for the population average startup costs \(\mu\) for candy store franchises.

Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below \(6 \mathrm{mg} / \mathrm{dl}(\) Reference: Manual of Laboratory and Diagnostic Tests, F. Fischbach). Recently, the patient's total calcium tests gave the following readings (in \(\mathrm{mg} / \mathrm{dl}\) ). \(9.3\) \(\begin{array}{llllll}8.8 & 10.1 & 8.9 & 9.4 & 9.8 & 10.0\end{array}\) \(\begin{array}{lll}9.9 & 11.2 & 12.1\end{array}\) (a) Use a calculator to verify that \(\bar{x}=9.95\) and \(s \approx 1.02\). (b) Find a \(99.9 \%\) confidence interval for the population mean of total calcium in this patient's blood. (c) Based on your results in part (b), do you think this patient still has a calcium deficiency? Explain.

A random sample of medical files is used to estimate the proportion \(p\) of all people who have blood type \(B\). (a) If you have no preliminary estimate for \(p\), how many medical files should you include in a random sample in order to be \(85 \%\) sure that the point estimate \(\hat{p}\) will be within a distance of \(0.05\) from \(p ?\) (b) Answer part (a) if you use the preliminary estimate that about 8 out of 90 people have blood type B. (Reference: Manual of Laboratory and Diagnostic Tests, F. Fischbach.)

Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. Forbes Top Companies, edited by J. T. Davis (John Wiley \& Sons), gave the following data about annual profits per employee (in units of one thousand dollars per employee) for representative companies in financial services. Companies such as Wells Fargo, First Bank System, and Key Banks were included. Assume \(\sigma \approx 10.2\) thousand dollars. $$ \begin{array}{llllllllll} 42.9 & 43.8 & 48.2 & 60.6 & 54.9 & 55.1 & 52.9 & 54.9 & 42.5 & 33.0 & 33.6 \\ 36.9 & 27.0 & 47.1 & 33.8 & 28.1 & 28.5 & 29.1 & 36.5 & 36.1 & 26.9 & 27.8 \\ 28.8 & 29.3 & 31.5 & 31.7 & 31.1 & 38.0 & 32.0 & 31.7 & 32.9 & 23.1 & 54.9 \\ 43.8 & 36.9 & 31.9 & 25.5 & 23.2 & 29.8 & 22.3 & 26.5 & 26.7 & & \end{array} $$ (a) Use a calculator or appropriate computer software to verify that, for the preceding data, \(\bar{x} \approx 36.0\). (b) Let us say that the preceding data are representative of the entire sector of (successful) financial services corporations. Find a \(75 \%\) confidence interval for \(\mu\), the average annual profit per employee for all successful banks. (c)Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (b). (d) Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b). (e) Repeat parts (b), (c), and (d) for a \(90 \%\) confidence level.

The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages (Reference: The Baseball Encyclopedia, Macmillan). $$ \begin{array}{llllllllll} 1.6 & 2.4 & 1.2 & 6.6 & 2.3 & 0.0 & 1.8 & 2.5 & 6.5 & 1.8 \\ 2.7 & 2.0 & 1.9 & 1.3 & 2.7 & 1.7 & 1.3 & 2.1 & 2.8 & 1.4 \\ 3.8 & 2.1 & 3.4 & 1.3 & 1.5 & 2.9 & 2.6 & 0.0 & 4.1 & 2.9 \\ 1.9 & 2.4 & 0.0 & 1.8 & 3.1 & 3.8 & 3.2 & 1.6 & 4.2 & 0.0 \\ 1.2 & 1.8 & 2.4 & & & & & & & \end{array} $$ (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x} \approx 2.29\) and \(s \approx 1.40 .\) (b) Compute a \(90 \%\) confidence interval for the population mean \(\mu\) of home run percentages for all professional baseball players. Hint: If you use Table 6 of Appendix II, be sure to use the closest \(d\). \(f\). that is smaller. (c) Compute a \(99 \%\) confidence interval for the population mean \(\mu\) of home run percentages for all professional baseball players. (d) The home run percentages for three professional players are Tim Huelett, \(2.5 \quad\) Herb Hunter, \(2.0 \quad\) Jackie Jensen, \(3.8\) Examine your confidence intervals and describe how the home run percentages for these players compare to the population average. (e) In previous problems, we assumed the \(x\) distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: See the central limit theorem in Section \(7.2 .\)

When \(\sigma\) is unknown and the sample is of size \(n \geq 30\), there are two methods for computing confidence intervals for \(\mu\). Method 1: Use the Student's \(t\) distribution with \(d . f .=n-1 .\) This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When \(n \geq 30\), use the sample standard deviation \(s\) as an estimate for \(\sigma\), and then use the standard normal distribution. This method is based on the fact that for large samples, \(s\) is a fairly good approximation for \(\sigma .\) Also, for large \(n\), the critical values for the Student's \(t\) distribution approach those of the standard normal distribution. Consider a random sample of size \(n=31\), with sample mean \(\bar{x}=45.2\) and sample standard deviation \(s=5.3\). (a) Compute \(90 \%, 95 \%\), and \(99 \%\) confidence intervals for \(\mu\) using Method 1 with a Student's \(t\) distribution. Round endpoints to two digits after the decimal. (b) Compute \(90 \%, 95 \%\), and \(99 \%\) confidence intervals for \(\mu\) using Method 2 with the standard normal distribution. Use \(s\) as an estimate for \(\sigma\). Round endpoints to two digits after the decimal. (c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's \(t\) distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution? (d) Repeat parts (a) through (c) for a sample of size \(n=81\). With increased sample size, do the two methods give respective confidence intervals that are more similar?

How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds, by Wirth and Young (Random House), claims that the air in the crown should be an average of \(100^{\circ} \mathrm{C}\) for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly \(100^{\circ} \mathrm{C}\). What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 56 readings (for a balloon in equilibrium) gave a mean temperature of \(\bar{x}=97^{\circ} \mathrm{C}\). For this balloon, \(\sigma \approx 17^{\circ} \mathrm{C}\). (a) Compute a \(95 \%\) confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (b) If the average temperature in the crown of the balloon goes above the high end of your confidence interval, do you expect that the balloon will go up or down? Explain.