91Ó°ÊÓ

Answer true or false. Explain your answer. For the same random sample, when the confidence level \(c\) is reduced, the confidence interval for \(\mu\) becomes shorter.

Archaeology: Ireland Inorganic phosphorous is a naturally occurring element in all plants and animals, with concentrations increasing progressively up the food chain (fruit \(<\) vegetables \(<\) cereals \(<\) nuts \(<\) corpse). Geochemical surveys take soil samples to determine phosphorous content (in ppm, parts per million). A high phosphorous content may or may not indicate an ancient burial site, food storage site, or even a garbage dump. The Hill of Tara is a very important archaeological site in Ireland. It is by legend the seat of Ireland's ancient high kings (Reference: Tara, An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Independent random samples from two regions in Tara gave the following phosphorous measurements (in ppm). Assume the population distributions of phosphorous are mound-shaped and symmetric for these two regions. $$ \begin{aligned} &\text { Region I: } x_{1} ; n_{1}=12\\\ &\begin{array}{llllll} 540 & 810 & 790 & 790 & 340 & 800 \\ 890 & 860 & 820 & 640 & 970 & 720 \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Region II: } x_{2} ; n_{2}=16\\\ &\begin{array}{llllllll} 750 & 870 & 700 & 810 & 965 & 350 & 895 & 850 \\ 635 & 955 & 710 & 890 & 520 & 650 & 280 & 993 \end{array} \end{aligned} $$ (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x}_{1} \approx 747.5, s_{1} \approx 170.4, \bar{x}_{2} \approx 738.9\), and \(s_{2} \approx 212.1 .\) (b) Let \(\mu_{1}\) be the population mean for \(x_{1}\) and let \(\mu_{2}\) be the population mean for \(x_{2} .\) Find a \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). (c) Interpretation Explain what the confidence interval means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the \(90 \%\) level of confidence, is one region more interesting than the other from a geochemical perspective? (d) Check Requirements Which distribution (standard normal or Student's \(t\). did you use? Why?

Large U.S. Companies: Foreign Revenue For large U.S. companies, what percentage of their total income comes from foreign sales? A random sample of technology companies (IBM, Hewlett-Packard, Intel, and others) gave the following information. $$ \begin{aligned} &\text { Technology companies, } \% \text { foreign revenue: } x_{1} ; n_{1}=16\\\ &\begin{array}{llllllll} 62.8 & 55.7 & 47.0 & 59.6 & 55.3 & 41.0 & 65.1 & 51.1 \\ 53.4 & 50.8 & 48.5 & 44.6 & 49.4 & 61.2 & 39.3 & 41.8 \end{array} \end{aligned} $$ Another independent random sample of basic consumer product companies (Goodyear, Sarah Lee, H.J. Heinz, Toys " \(q\) "Us) gave the following information. $$ \begin{aligned} &\text { Basic consumer product companies, } \% \text { foreign revenue: } x_{2} ; n_{2}=17\\\ &\begin{array}{llllllll} 28.0 & 30.5 & 34.2 & 50.3 & 11.1 & 28.8 & 40.0 & 44.9 \\ 40.7 & 60.1 & 23.1 & 21.3 & 42.8 & 18.0 & 36.9 & 28.0 \end{array}\\\ &\begin{aligned} &40.7 \\ &32.5 \end{aligned} \end{aligned} $$ (Reference: Forbes Top Companies.) Assume that the distributions of percentage foreign revenue are mound-shaped and symmetric for these two company types. (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x}_{1} \approx 51.66, s_{1} \approx 7.93, \bar{x}_{2} \approx 33.60\), and \(s_{2} \approx 12.26\). (b) Let \(\mu_{1}\) be the population mean for \(x_{1}\) and let \(\mu_{2}\) be the population mean for \(x_{2}\). Find an \(85 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). (c) Interpretation Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the \(85 \%\) level of confidence, do technology companies have a greater percentage foreign revenue than basic consumer product companies? (d) Check Requirements Which distribution (standard normal or Student's \(t)\) did you use? Why?

Assume that the population of \(x\) values has an approximately normal distribution. Archaeology: Tree Rings At Burnt Mesa Pueblo, the method of tree-ring dating gave the following years A.D. for an archaeological excavation site (Bandelier Archaeological Excavation Project: Summer 1990 Excavations at Burnt Mesa Pueblo, edited by Kohler, Washington State University): \(\begin{array}{lllllllll}1189 & 1271 & 1267 & 1272 & 1268 & 1316 & 1275 & 1317 & 1275\end{array}\) (a) Use a calculator with mean and standard deviation keys to verify that the sample mean year is \(\bar{x} \approx 1272\), with sample standard deviation \(s \approx 37\) years. (b) Find a \(90 \%\) confidence interval for the mean of all tree-ring dates from this archaeological site. (c) Interpretation What does the confidence interval mean in the context of this problem?

Assume that the population of \(x\) values has an approximately normal distribution. Franchise: Candy Store Do you want to own your own candy store? 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 ). \(\begin{array}{lllllllll}95 & 173 & 129 & 95 & 75 & 94 & 116 & 100 & 85\end{array}\) (a) 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. (b) Find a \(90 \%\) confidence interval for the population average startup costs \(\mu\) for candy store franchises. (c) Interpretation What does the confidence interval mean in the context of this problem?

Small Business: Bankruptcy The National Council of Small Businesses is interested in the proportion of small businesses that declared Chapter 11 bankruptcy last year. Since there are so many small businesses, the National Council intends to estimate the proportion from a random sample. Let \(p\) be the proportion of small businesses that declared Chapter 11 bankruptcy last year. (a) If no preliminary sample is taken to estimate \(p\), how large a sample is necessary to be \(95 \%\) sure that a point estimate \(\hat{p}\) will be within a distance of \(0.10\) from \(p\) ? (b) In a preliminary random sample of 38 small businesses, it was found that six had declared Chapter 11 bankruptcy. How many more small businesses should be included in the sample to be \(95 \%\) sure that a point estimate \(\hat{p}\) will be within a distance of \(0.10\) from \(p\) ?