Stats: Data and Models 4th Edition solutions

Stats: Data and Models 4
Residuals Suppose you have fit a linear model to some data and now take a look at the residuals. For each of the following possible residuals plots, tell whether you would try a re-expression and, if so, why.

Stats: Data and Models 4
Problem 1RE College Every year, US News and World Report publishes a special issue on many U.S. colleges and universities. The scatterplots below have Student/Faculty Ratio (number of students per faculty member) for the colleges and universities on the y-axes plotted against 4 other variables. The c

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Problem 2E College Every year US News and World Report publishes a special issue on many U.S. colleges and universities. The scatterplots below have Student/Faculty Ratio (number of students per faculty member) for the colleges and universities on the y-axes plotted against 4 other variables. The cor

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Problem 2RE How old is that tree? One can determine how old a tree is by counting its rings, but that requires cutting the tree down. Can we estimate the trees age simply from its diameter? A forester measured 27 trees of the same species that had been cut down, and counted the rings to determine the

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Vineyards, more information Here are the scatterplot and regression analysis for Case Prices of 36 wines from vineyards in the Finger Lakes region of New York State and the Ages of the vineyards. (Data in Vineyards full): a) Does it appear that vineyards in business longer get highe

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Problem 4RE Vineyards again Instead of Age, perhaps the Size of the vineyard (in acres) is associated with the price of the wines. Look at the scatterplot (Data in Vineyards full): a) Do you see any evidence of an association? b) What concern do you have about this scatterplot? c) If the red + da

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Problem 7RE Acid rain Biologists studying the effects of acid rain on wildlife collected data from 163 streams in the Adirondack Mountains. They recorded the pH (acidity) of the water and the BCI, a measure of biological diversity, and they calculated R2 = 27%. Heres a scatterplot of BCI against pH:

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Problem 9E Models For each of the models listed below, predict y when x = 2.

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More models For each of the models listed below, predict y when x = 2. a. \(\hat{y}=1.2+0.8 \log x\) b. \(\log \hat{y}=1.2+0.8 x\) c. \(\ln \hat{y}=1.2+0.8 \ln x\) d. \(\hat{y}^{2}=1.2+0.8 x\) e. \(\frac{1}{\sqrt{\hat{y}}}=1.2+0.8 x\)

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Problem 10RE Grades A Statistics instructor created a linear regression equation to predict students final exam scores from their midterm exam scores. The regression equation was a) If Susan scored a 70 on the midterm, what did the instructor predict for her score on the final? b) Susan got an 80

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Problem 11E Vineyards again Instead of Age, perhaps the Size of the vineyard (in acres) is associated with the price of the wines. Look at the scatterplot: a) Do you see any evidence of an association? ________________ b) What concern do you have about this scatterplot? ________________ c) If

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Problem 11RE Tips Its commonly believed that people use tips to reward good service. A researcher for the hospitality industry examined tips and ratings of service quality from 2645 dining parties at 21 different restaurants. The correlation between ratings of service and tip percentages was 0.11. (M

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Problem 12E Crowdedness In a Chance magazine article (Summer 2005), Danielle Vasilescu and Howard Wainer used data from the United Nations Center for Human Settlements to investigate aspects of living conditions for several countries. Among the variables they looked at were the countrys per capita gr

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Problem 12RE Cramming One Thursday, researchers gave students enrolled in a section of basic Spanish a set of 50 new vocabulary words to memorize. On Friday, the students took a vocabulary test. When they returned to class the following Monday, they were retestedwithout advance warning. Here are the

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Gas mileage, revisited Lets try the re-expressed variable Fuel Consumption (gal/100 mi) to examine the fuel efficiency of the 11 cars in Exercise 11. Here are the revised regression analysis and residuals plot: a) Explain why this model appears to be better than the linear m

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Problem 13RE Logs (not logarithms) The value of a log is based on the number of board feet of lumber the log may contain. (A board foot is the equivalent of a piece of wood 1 inch thick, 12 inches wide, and 1 foot long. For example, a 2" * 4" piece that is 12 feet long contains 8 board feet.) To esti

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Crowdedness again In Exercise 12 we looked at United Nations data about a countrys GDP and the average number of people per room (Crowdedness) in housing there. For a re-expression, a student tried the reciprocal -10000/GDP, representing the number of people per $10,000 of gross domestic product

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Problem 14RE US cities Data from 50 large U.S. cities show the mean January Temperature and the Latitude. Describe what you see in the scatterplot.

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Problem 15RE Correlations The study of U.S. cities in Exercise found the mean January Temperature (degrees Fahrenheit), Altitude (feet above sea level), and Latitude (degrees north of the equator) for 55 cities. Heres the correlation matrix: Jan. Temp Latitude Altitude Jan. Temp 1.000

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Colorblind Although some women are colorblind, this condition is found primarily in men. Why is it wrong to say theres a strong correlation between Sex and Colorblindness?

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Problem 17RE Life expectancy The data in the table below list the Life Expectancy for white males in the United States every decade during the last century (1 = 1900 to 1910, 2 = 1911 to 1920, etc.). Create a model to predict future increases in life expectancy. (National Vital Statistics Report) Dec

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Problem 18E Crowdedness In a Chance magazine article (Summer 2005), Danielle Vasilescu and Howard Wainer used data from the United Nations Center for Human Settlements to investigate aspects of living conditions for several countries. Among the variables they looked at were the countrys per capita gr

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Problem 19E Gas mileage revisited Lets try the re-expressed variable Fuel Consumption 1gal>100 mi2 to examine the fuel efficiency of the 11 cars in Exercise. Here are the revised regression analysis and residuals plot: Dependent variable is: Fuel Consumption R-squared = 89.2% Variable Coefficient Int

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How old is that tree? One can determine how old a tree is by counting its rings, but that requires either cutting the tree down or extracting a sample from the trees core. Can we estimate the trees age simply from its diameter? A forester measured 27 trees of the same species that had been cut d

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Problem 20E Pendulum A student experimenting with a pendulum counted the number of full swings the pendulum made in 20 seconds for various lengths of string. Her data are shown below. a) Explain why a linear model is not appropriate for using the Length of a pendulum to predict the Number of Swin

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Problem 20RE Improving trees In the last exercise, you saw that the linear model had some deficiencies. Lets create a better model. a) Perhaps the cross-sectional area of a tree would be a better predictor of its age. Since area is measured in square units, try re-expressing the data by squaring the

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Problem 21RE Big screen An electronics website collects data on the size of new HD flat panel televisions (measuring the diagonal of the screen in inches) to predict the cost (in hundreds of dollars). Which of these is most likely to be the slope of the regression line: 0.03, 0.3, 3, 30? Explain.

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Problem 22E Crowdedness again In Exercise we looked at United Nations data about a countrys GDP and the average number of people per room (Crowdedness) in housing there. For a re-expression, a student tried the reciprocal -10000>GDP, representing the number of people per $10,000 of gross domestic pro

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Problem 23E Planets, distances and order Lets look again at the pattern in the locations of the planets in our solar system seen in the table in Exercise 22. a) Re-express the distances to create a model for the Distance from the sun based on the planets Position. b) Based on this model, would you ag

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Problem 24E Traffic Highway planners investigated the relationship between traffic Density (number of automobiles per mile) and the average Speed of the traffic on a moderately large city thoroughfare. The data were collected at the same location at 10 different times over a span of 3 months. They fo

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Problem 24RE Orange production The table below shows that as the number of oranges on a tree increases, the fruit tends to get smaller. Create a model for this relationship, and express any concerns you may have. Number of Oranges/Tree Average Weight/Fruit(lb) 50 0.60 100 0.58 150 0.56 200 0

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Problem 25E Cramming One Thursday, researchers gave students enrolled in a section of basic Spanish a set of 50 new vocabulary words to memorize. On Friday the students took a vocabulary test. When they returned to class the following Monday, they were retestedwithout advance warning. Here are the te

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Problem 25RE U.S. cities Data from 50 large U.S. cities show the mean January Temperature and the Latitude. Describe what you see in the scatterplot.

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Planets, models, and laws The model you found in Exercise 22 is a relationship noted in the 17th century by Kepler as his Third Law of Planetary Motion. It was subsequently explained as a consequence of Newtons Law of Gravitation. The models for Exercises 2325 relate to what is sometimes called

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Problem 26RE French Consider the association between a students score on a French vocabulary test and the weight of the student. What direction and strength of correlation would you expect in each of the following situations? Explain. a) The students are all in third grade. ________________ b) The

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Problem 27E GDP The scatterplot shows the gross domestic product (GDP) of the United States in billions of dollars plotted against years since 1950. A linear model fit to the relationship looks like this: Dependent variable is: GDP R-squared = 87.6% s = 1597.7456 Variable Coefficient Intercept -2

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Problem 27RE Twins Twins are often born after a pregnancy that lasts less than 9 months. On the next page is a graph from the Journal of the American Medical Association (JAMA) showing the rate of preterm twin births in the United States over the past 20 years. In this study, JAMA categorized mothers

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Problem 28RE Tree growth A 1996 study examined the growth of grapefruit trees in Texas, determining the average trunk Diameter (in inches) for trees of varying Ages: Age (yr) 2 4 6 8 10 12 14 16 18 20 Diameter (in.) 2.1 3.9 5.2 6.2 6.9 7.6 8.3 9.1 10.0 11.4 a) Fit a linear model

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Slower is cheaper? Researchers studying how a cars Fuel Efficiency varies with its Speed drove a compact car 200 miles at various speeds on a test track. Their data are shown in the table.

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Problem 31RE Gasoline Since clean-air regulations have dictated the use of unleaded gasoline, the supply of leaded gas in New York state has diminished. The table below was given on the August 2001 New York State Math B exam, a statewide achievement test for high school students. Year 1984 1988 19

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Orange production The table below shows that as the number of oranges on a tree increases, the fruit tends to get smaller. Create a model for this relationship, and express any concerns you may have.

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Lunchtime Does how long toddlers sit at the lunch table help predict how much they eat? The table and graph show the number of minutes the kids stayed at the table and the number of calories they consumed. Create and interpret a model for these data.

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Problem 34E Pressure Scientist Robert Boyle examined the relationship between the volume in which a gas is contained and the pressure in its container. He used a cylindrical container with a moveable top that could be raised or lowered to change the volume. He measured the Height in inches by countin

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Problem 34RE Tobacco and alcohol Are people who use tobacco products more likely to consume alcohol? Here are data on household spending (in pounds) taken by the British Government on 11 regions in Great Britain. Do tobacco and alcohol spending appear to be related? What questions do you have about t

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Tobacco and alcohol Are people who use tobacco products more likely to consume alcohol? Here are data on household spending (in pounds) taken by the British government on 11 regions in Great Britain. Do tobacco and alcohol spending appear to be related? What questions do you have about these data? Wh

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Problem 36RE Football weights The Sears Cup was established in 1993 to honor institutions that maintain a broad-based athletic program, achieving success in many sports, both mens and womens. Since its Division III inception in 1995, the cup has been won by Williams College 15 of 17 years. Their foot

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Problem 37RE Models Find the predicted value of y, using each model for x = 10. a) ? = 2 + 0.8 ln x ________________ b) log ? = 5 - 0.23x ________________ c)

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Williams vs. Texas Here are the average weights of the football team for the University of Texas for various years in the 20th century. a) Fit a straight line to the relationship of Weight by Year for Texas football players. b) According to these models, in what year will the pr

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Problem 39RE Vehicle weights The Minnesota Department of Transportation hoped that they could measure the weights of big trucks without actually stopping the vehicles by using a newly developed weigh-in-motion scale. After installation of the scale, a study was conducted to find out whether the scale

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Problem 40RE Profit How are a companys profits related to its sales? Lets examine data from 71 large U.S. corporations. All amounts are in millions of dollars. a) Histograms of Profits and Sales and histograms of the logarithms of Profits and Sales appear below. Why are the re-expressed data better f

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Problem 41RE Down the drain Most water tanks have a drain plug so that the tank may be emptied when its to be moved or repaired. How long it takes a certain size of tank to drain depends on the size of the plug, as shown in the table. Create a model. Plug Dia (in.) ? 1 1 1 2 Drain Time (min

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Problem 42RE Chips A start-up company has developed an improved electronic chip for use in laboratory equipment. The company needs to project the manufacturing cost, so it develops a spreadsheet model that takes into account the purchase of production equipment, overhead, raw materials, depreciation,