2.16 Make some sketches. For each of the following situations, make a scatterplot that illustrates the given relationship between two variables.

2.17 What’s wrong? Explain what is wrong with each of the following:

2.18 Blueberries and anthocyanins. Anthocyanins are compounds that have been associated with health benefits associated with the heart, bones, and the brain. Blueberries are a good source of many different anthocyanins. Researchers at the Piedmont Research Station of North Carolina State University have assembled a database giving the concentrations of 18 different anthocyanins for 267 varieties of blueberries.⁹ Four of the anthocyanins measured are delphinidin-3-arabinoside, malvidin-3-arabinoside, cyanidin-3-galactoside, and delphinidin-3-glucoside, all measured in units of mg per 100g of berries (dry weight). In the data file, we have simplified the names of these anthocyanins to Antho1, Antho2, Antho3, and Antho4. In Exercises 1.167 and 1.168 (page 77), you examined the distributions of each Antho3 and Antho4.

2.19 Blueberries and anthocyanins with logs. Refer to the previous exercise. Transform each of the variables with a log, make a scatterplot and answer the questions in the previous exercise for the transformed data.

2.20 Blueberries and anthocyanins: Raw data or logs. Refer to Exercises 2.18 and 2.19.

2.21 Fuel consumption. Natural Resources Canada tests new vehicles each year and reports several variables related to fuel consumption for vehicles in different classes.¹⁰ For 2015 they provide data for 527 vehicles that use regular fuel. Two variables reported are carbon dioxide (CO₂) emissions and highway fuel consumption. CO₂ is measured in grams per kilometer (g/km) and highway fuel consumption measured in liters per 100 kilometers (L/100km).

2.22 Fuel consumption with a line. Refer to the previous exercise.

2.23 Fuel consumption for different types of vehicles. Refer to the previous two exercises. Those exercises examined data for vehicles that used regular fuel. Data are also available for vehicles that use several other types of fuel. There are 1067 vehicles in total. The variable Fuel has four different possible values: X, for regular fuel; Z, for premium fuel; D, for diesel; and E, for ethanol.

2.24 Bone strength. Osteoporosis is a condition where bones become weak. It affects more than 200 million people worldwide. Exercise is one way to produce strong bones and to prevent osteoporosis. Because we use our dominant arm (the right arm for most people) more than our nondominant arm, we expect the bone in our dominant arm to be stronger than the bone in our nondominant arm. By comparing the strengths, we can get an idea of the effect that exercise can have on bone strength. Here are some data on the strength of bones, measured in Newton meters divided by 1000 (Nm/1000), for the arms of 15 young men:¹¹

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Before attempting to compare the arm strengths of the nondominant and dominant arms, let’s take a careful look at the data for these two variables.

2.25 Bone strength for baseball players. Refer to the previous exercise. The study collected arm bone strength information for two groups of young men. The data in the previous exercise were for a control group. The second group in the study comprised men who played baseball. We know that these baseball players use their dominant arm in throwing (those who throw with their nondominant arm were excluded), so they get more arm exercise than the controls. Here are the data for the baseball players:

Answer the questions in the previous exercise for the baseball players.

2.26 Compare the baseball players with the controls. Refer to the previous two exercises.

2.27 Parents’ income and student loans. How well does the income of a college student’s parents predict how much the student will borrow to pay for college? We have data on parents’ income and college debt for a sample of 1200 recent college graduates. What are the explanatory and response variables? Are these variables categorical or quantitative? Do you expect a positive or negative association between these variables? Why?

2.28 What’s in the beer? The website beer100.com advertises itself as “Your Place for All Things Beer.” One of their “things” is a list of 159 domestic beer brands with the percent alcohol, calories per 12 ounces, and carbohydrates per 12 ounces (in grams).¹²

2.29 More beer. Refer to the previous exercise.

2.30 Imported beer. The beer100 website also gives data for imported beers. Describe the relationship between calories and percent alcohol for these imported beers.

2.31 Compare domestic with imported. Plot calories versus percent alcohol for domestic and imported beers on the same scatterplot. Use different colors or symbols for the two types of beers. Summarize what this plot tells you about the relationship and the difference between the two types of beer. In particular, note any characteristics that are better shown in this plot relative to what was learned in Exercises 2.28, 2.29, and 2.30.

2.32 Decay of a radioactive element. Barium-137m is a radioactive form of the element barium that decays very rapidly. It is easy and safe to use for lab experiments in schools and colleges.¹³ In a typical experiment, the radioactivity of a sample of barium-137m is measured for one minute. It is then measured for three additional one-minute periods, separated by two minutes. So data are recorded at one, three, five, and seven minutes after the start of the first counting period. The measurement units are counts. Here are the data for one of these experiments:¹⁴

2.33 Use a log for the radioactive decay. Refer to the previous exercise. Transform the counts using a log transformation. Then repeat parts (a) through (e) for the transformed data and compare your results with those from the previous exercise.

2.34 Internet use and babies. The World Bank collects data on many variables related to world development for countries throughout the world. Two of these are Internet use, in number of users per 100 people, and birthrate, in births per 1000 people.¹⁵ Figure 2.13 is a scatterplot of birthrate versus Internet use for the 106 countries that have data available for both variables.

2.35 Try a log. Refer to the previous exercise.

2.36 Make another plot. Refer to Exercise 2.34.

2.37 Body mass and metabolic rate. Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. The following table gives data on the lean body mass and resting metabolic rate for 12 women and 7 men who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person’s weight leaving out all fat. Metabolic rate is measured in calories burned per 24 hours, the same calories used to describe the energy content of foods. The researchers believe that lean body mass is an important influence on metabolic rate.