Chapter 6

Environmental Protection Agency (EPA) fuel economy estimates for automobile models tested recently predicted a mean of \(24.8 \mathrm{mpg}\) and a standard deviation of \(6.2 \mathrm{mpg}\) for highway driving. Assume that a Normal model can be applied. a) Draw the model for auto fuel economy. Clearly label it, showing what the \(68-95-99.7\) Rule predicts. b) In what interval would you expect the central \(68 \%\) of autos to be found? c) About what percent of autos should get more than \(31 \mathrm{mpg}\) ? d) About what percent of cars should get between 31 and \(37.2 \mathrm{mpg}\) ? e) Describe the gas mileage of the worst \(2.5 \%\) of all cars.

Some IQ tests are standardized to a Normal model, with a mean of 100 and a standard deviation of \(16 .\) a) Draw the model for these IQ scores. Clearly label it, showing what the \(68-95-99.7\) Rule predicts. b) In what interval would you expect the central \(95 \%\) of IQ scores to be found? c) About what percent of people should have IQ scores above \(116 ?\) d) About what percent of people should have IQ scores between 68 and 84 ? e) About what percent of people should have IQ scores above \(132 ?\)

In Exercise 17 we suggested the model \(N(1152,84)\) for weights in pounds of yearling Angus steers. What weight would you consider to be unusually low for such an animal? Explain.

. Exercise 26 proposes modeling IQ scores with \(N(100,16) .\) What IQ would you consider to be unusually high? Explain.

A forester measured 27 of the trees in a large woods that is up for sale. He found a mean diameter of \(10.4\) inches and a standard deviation of \(4.7\) inches. Suppose that these trees provide an accurate description of the whole forest and that a Normal model applies. a) Draw the Normal model for tree diameters. b) What size would you expect the central \(95 \%\) of all trees to be? c) About what percent of the trees should be less than an inch in diameter? d) About what percent of the trees should be between \(5.7\) and \(10.4\) inches in diameter? e) About what percent of the trees should be over 15 inches in diameter?

A company that manufactures rivets believes the shear strength (in pounds) is modeled by \(N(800,50)\). a) Draw and label the Normal model. b) Would it be safe to use these rivets in a situation requiring a shear strength of 750 pounds? Explain. c) About what percent of these rivets would you expect to fall below 900 pounds? d) Rivets are used in a variety of applications with varying shear strength requirements. What is the maximum shear strength for which you would feel comfortable approving this company's rivets? Explain your reasoning.

The mean of the 100 car speeds in Exercise 20 was \(23.84 \mathrm{mph}\), with a standard deviation of \(3.56 \mathrm{mph}\). a) Using a Normal model, what values should border the middle \(95 \%\) of all car speeds? b) Here are some summary statistics. $$ \begin{array}{lll} \hline \text { Percentile } & & \text { Speed } \\ \hline 100 \% & \text { Max } & 34.060 \\ 97.5 \% & & 30.976 \\ 90.0 \% & & 28.978 \\ 75.0 \% & \text { Q3 } & 25.785 \\ 50.0 \% & \text { Median } & 23.525 \\ 25.0 \% & \text { Q1 } & 21.547 \\ 10.0 \% & & 19.163 \\ 2.5 \% & & 16.638 \\ 0.0 \% & \text { Min } & 16.270 \\ \hline \end{array} $$

NFL data from the 2006 football season reported the number of yards gained by each of the league's 167 wide receivers: a) According to the Normal model, what percent of receivers would you expect to gain fewer yards than 2 standard deviations below the mean number of yards? b) For these data, what does that mean? c) Explain the problem in using a Normal model here.

Using \(N(1152,84)\), the Normal model for weights of Angus steers in Exercise 17, what percent of steers weigh a) over 1250 pounds? b) under 1200 pounds? c) between 1000 and 1100 pounds?

Based on the Normal model \(N(100,16)\) describing IQ scores, what percent of people's IQs would you expect to be a) over \(80 ?\) b) under \(90 ?\) c) between 112 and \(132 ?\)