SECTION 2.3 Exercises

For Exercises 2.44 and 2.45, see page 82; for 2.46, see page 84; for 2.47, see page 86; for 2.48 and 2.49, see page 88; for 2.50, see page 90; for 2.51, see page 90; for 2.52, see page 91; and for 2.53, see page 94.

Question 2.54

2.54 What is the equation for the selling price?

You buy items at a cost of and sell them for . Assume that your selling price includes a profit of 12% plus a fixed cost of $25.00. Give an equation that can be used to determine from .

Question 2.55

2.55 Production costs for cell phone batteries

A company manufactures batteries for cell phones. The overhead expenses of keeping the factory operational for a month—even if no batteries are made—total $500,000. Batteries are manufactured in lots (1000 batteries per lot) costing $7000 to make. In this scenario, $500,000 is the fixed cost associated with producing cell phone batteries and $7000 is the marginal (or variable) cost of producing each lot of batteries. The total monthly cost of producing lots of cell phone batteries is given by the equation

  1. Draw a graph of this equation. (Choose two values of , such as 0 and 20, to draw the line and a third for a check. Compute the corresponding values of from the equation. Plot these two points on graph paper and draw the straight line joining them.)
  2. What will it cost to produce 15 lots of batteries (15,000 batteries)?
  3. If each lot cost $10,000 instead of $7000 to produce, what is the equation that describes total monthly cost for lots produced?

2.55

(b) For , . (c) .

Question 2.56

2.56 Inventory of Blu-Ray players

A local consumer electronics store sells exactly eight Blu-Ray players of a particular model each week. The store expects no more shipments of this particular model, and they have 96 such units in their current inventory.

  1. Give an equation for the number of Blu-Ray players of this particular model in inventory after x weeks. What is the slope of this line?
  2. Draw a graph of this line between now (Week 0) and Week 10.
  3. Would you be willing to use this line to predict the inventory after 25 weeks? Do the prediction and think about the reasonableness of the result.

Question 2.57

2.57 Compare the cell phone payment plans

A cellular telephone company offers two plans. Plan A charges $30 a month for up to 120 minutes of airtime and $0.55 per minute above 120 minutes. Plan B charges $35 a month for up to 200 minutes and $0.50 per minute above 200 minutes.

  1. Draw a graph of the Plan A charge against minutes used from 0 to 250 minutes.
  2. How many minutes a month must the user talk in order for Plan B to be less expensive than Plan A?

2.57

(b) Plan B gets cheaper at 130 minutes.

Question 2.58

2.58 Companies of the world

Refer to Exercise 1.118 (page 61), where we examined data collected by the World Bank on the numbers of companies that are incorporated and listed on their country's stock exchange at the end of the year. In Exercise 2.10, you examined the relationship between these numbers for 2012 and 2002, and in Exercise 2.27, you found the correlation between these two variables.

inccom

  1. Find the least-squares regression equation for predicting the 2012 numbers using the 2002 numbers.
  2. Sweden had 332 companies in 2012 and 278 companies in 2002. Use the least-squares regression equation to find the predicted number of companies in 2012 for Sweden.
  3. Find the residual for Sweden.

96

Question 2.59

2.59 Companies of the world

Refer to the previous exercise and to Exercise 2.11 (page 72). Answer parts (a), (b), and (c) of the previous exercise for 2012 and 1992. Compare the results you found in the previous exercise with the ones you found in this exercise. Explain your findings in a short paragraph.

inccom

2.59

(a) . (b) For , . (c) .

Question 2.60

2.60 A product for lab experiments

In Exercise 2.17 (page 73), you described the relationship between time and count for an experiment examining the decay of barium. In Exercise 2.29 (page 78), you found the correlation between these two variables.

decay

  1. Find the least-squares regression equation for predicting count from time.
  2. Use the equation to predict the count at one, three, five, and seven minutes.
  3. Find the residuals for one, three, five, and seven minutes.
  4. Plot the residuals versus time.
  5. What does this plot tell you about the model you used to describe this relationship?

Question 2.61

2.61 Use a log for the radioactive decay

Refer to the previous exercise. Also see Exercise 2.18 (page 73), where you transformed the counts with a logarithm, and Exercise 2.30 (pages 7879), where you found the correlation between time and the log of the counts. Answer parts (a) to (e) of the previous exercise for the transformed counts and compare the results with those you found in the previous exercise.

decay

2.61

(a) . (b) For , . For , . For , . For , . (c) For , . For , . For , . For , .

Question 2.62

2.62 Fuel efficiency and CO2 emissions

In Exercise 2.37 (page 79), you examined the relationship between highway MPG and city MPG for 1067 vehicles for the model year 2014.

canfuel

  1. Use the city MPG to predict the highway MPG. Give the equation of the least-squares regression line.
  2. The Lexus 350h AWD gets 42 MPG for city driving and 38 MPG for highway driving. Use your equation to find the predicted highway MPG for this vehicle.
  3. Find the residual.

Question 2.63

2.63 Fuel efficiency and CO2 emissions

Refer to the previous exercise.

canfuel

  1. Make a scatterplot of the data with highway MPG as the response variable and city MPG as the explanatory variable. Include the least-squares regression line on the plot. There is an unusual pattern for the vehicles with high city MPG. Describe it.
  2. Make a plot of the residuals versus city MPG. Describe the major features of this plot. How does the unusual pattern noted in part (a) appear in this plot?
  3. The Lexus 350h AWD that you examined in parts (b) and (c) of the previous exercise is in the group of unusual cases mentioned in parts (a) and (b) of this exercise. It is a hybrid vehicle that uses a conventional engine and a electric motor that is powered by a battery that can recharge when the vehicle is driven. The conventional engine also turns off when the vehicle is stopped in traffic. As a result of these features, hybrid vehicles are unusually efficient for city driving, but they do not have a similar advantage when driven at higher speeds on the highway. How do these facts explain the residual for this vehicle?
  4. Several Toyota vehicles are also hybrids. Use the residuals to suggest which vehicles are in this category.

2.63

(a) The vehicles with high city MPG don't follow the regression line; rather, they have a much lower highway MPG than the regression line would predict. (b) For the vehicles with high city MPG, all of the residuals are negative, creating a curve in the plot, suggesting a possible transformation is necessary. (c) Because the hybrid vehicles have an electric motor in addition to the conventional motor, which is intended to improve city MPG, we would expect them to have a much better city MPG than expected, which is why their residuals fall so far below the residuals for the conventional motor vehicles. (d) Three Toyota Prius models and two Toyota Camry Hybrid models likely are hybrids.

Question 2.64

2.64 Consider the fuel type

Refer to the previous two exercises and to Figure 2.6 (page 71), where different colors are used to distinguish four different types of fuels used by these vehicles. In Exercise 2.38, you examined the relationship between Highway MPG and City MPG for each of the four different fuel types used by these vehicles. Using the previous two exercises as a guide, analyze these data separately for each of the four fuel types. Write a summary of your findings.

canfuel

Question 2.65

2.65 Predict one characteristic of a product using another characteristic

In Exercise 2.12 (page 72), you used a scatterplot to examine the relationship between calories per 12 ounces and percent alcohol in 175 domestic brands of beer. In Exercise 2.31 (page 79), you calculated the correlation between these two variables.

beer

  1. Find the equation of the least-squares regression line for these data.
  2. Make a scatterplot of the data with the least-squares regression line.

2.65

(a) .

Question 2.66

2.66 Predicted values and residuals

Refer to the previous exercise.

beer

  1. New Belgium Fat Tire is 5.2 percent alcohol and has 160 calories per 12 ounces. Find the predicted calories for New Belgium Fat Tire.
  2. Find the residual for New Belgium Fat Tire.

97

Question 2.67

2.67 Predicted values and residuals

Refer to the previous two exercises.

beer

  1. Make a plot of the residuals versus percent alcohol.
  2. Interpret the plot. Is there any systematic pattern? Explain your answer.
  3. Examine the plot carefully and determine the approximate location of New Belgium Fat Tire. Is there anything unusual about this case? Explain why or why not.

2.67

(b) The residual plot looks fairly random. (c) There is nothing unusual about the location of New Belgium Fat Tire.

Question 2.68

2.68 Carbohydrates and alcohol in beer revisited

Refer to Exercise 2.65. The data that you used to compute the least-squares regression line includes a beer with a very low alcohol content that might be considered to be an outlier.

beer

  1. Remove this case and recompute the least-squares regression line.
  2. Make a graph of the regression lines with and without this case.
  3. Do you think that this case is influential? Explain your answer.

Question 2.69

2.69 Monitoring the water quality near a manufacturing plant

Manufacturing companies (and the Environmental Protection Agency) monitor the quality of the water near manufacturing plants. Measurements of pollutants in water are indirect—a typical analysis involves forming a dye by a chemical reaction with the dissolved pollutant, then passing light through the solution and measuring its “absorbance.” To calibrate such measurements, the laboratory measures known standard solutions and uses regression to relate absorbance to pollutant concentration. This is usually done every day. Here is one series of data on the absorbance for different levels of nitrates. Nitrates are measured in milligrams per liter of water.10

nrates

Nitrates Absorbance Nitrates Absorbance
50 7.0 800 93.0
50 7.5 1200 138.0
100 12.8 1600 183.0
200 24.0 2000 230.0
400 47.0 2000 226.0
  1. Chemical theory says that these data should lie on a straight line. If the correlation is not at least 0.997, something went wrong and the calibration procedure is repeated. Plot the data and find the correlation. Must the calibration be done again?
  2. What is the equation of the least-squares line for predicting absorbance from concentration? If the lab analyzed a specimen with 500 milligrams of nitrates per liter, what do you expect the absorbance to be? Based on your plot and the correlation, do you expect your predicted absorbance to be very accurate?

2.69

(a) The correlation is 0.9999, so the calibration does not need to be repeated. (b) . For , . Because the relationship is so strong, , we would expect our predicted absorbance to be very accurate.

Question 2.70

2.70 Data generated by software

The following 20 observations on and were generated by a computer program.

gendata

34.38 22.06 27.07 17.75
30.38 19.88 31.17 19.96
26.13 18.83 27.74 17.87
31.85 22.09 30.01 20.20
26.77 17.19 29.61 20.65
29.00 20.72 31.78 20.32
28.92 18.10 32.93 21.37
26.30 18.01 30.29 17.31
29.49 18.69 28.57 23.50
31.36 18.05 29.80 22.02
  1. Make a scatterplot and describe the relationship between and .
  2. Find the equation of the least-squares regression line and add the line to your plot.
  3. Plot the residuals versus .
  4. What percent of the variability in is explained by ?
  5. Summarize your analysis of these data in a short paragraph.

Question 2.71

2.71 Add an outlier

Refer to the previous exercise. Add an additional case with and to the data set. Repeat the analysis that you performed in the previous exercise and summarize your results, paying particular attention to the effect of this outlier.

gendatb

2.71

(a) There seems to be a weak positive linear relationship between and , but with one extreme outlier with a very high value. (b) . (d) (e) Although the variable accounts for 77.15% of the variation in , there is a very high outlier for , which pulls the regression line unnaturally. This is seen in the jump of the -square value from 27% up to 77%, indicating that this observation is very influential in the analysis. This is also demonstrated by the systematic pattern in both the scatterplot and the residual plot, with most of the data points forming a line except for the outlier.

Question 2.72

2.72 Add a different outlier

Refer to the previous two exercises. Add an additional case with and to the original data set.

gendatc

  1. Repeat the analysis that you performed in the first exercise and summarize your results, paying particular attention to the effect of this outlier.
  2. In this exercise and in the previous one, you added an outlier to the original data set and reanalyzed the data. Write a short summary of the changes in correlations that can result from different kinds of outliers.

Question 2.73

2.73 Influence on correlation

The Correlation and Regression applet at the text website allows you to create a scatterplot and to move points by dragging with the mouse. Click to create a group of 12 points in the lower-left corner of the scatterplot with a strong straight-line pattern (correlation about 0.9).

98

  1. Add one point at the upper right that is in line with the first 12. How does the correlation change?
  2. Drag this last point down until it is opposite the group of 12 points. How small can you make the correlation? Can you make the correlation negative? You see that a single outlier can greatly strengthen or weaken a correlation. Always plot your data to check for outlying points.

Question 2.74

2.74 Influence in regression

As in the previous exercise, create a group of 12 points in the lower-left corner of the scatterplot with a strong straight-line pattern (correlation at least 0.9). Click the “Show least-squares line” box to display the regression line.

  1. Add one point at the upper right that is far from the other 12 points but exactly on the regression line. Why does this outlier have no effect on the line even though it changes the correlation?
  2. Now drag this last point down until it is opposite the group of 12 points. You see that one end of the least-squares line chases this single point, while the other end remains near the middle of the original group of 12. What about the last point makes it so influential?

Question 2.75

2.75 Employee absenteeism and raises

Data on number of days of work missed and annual salary increase for a company's employees show that, in general, employees who missed more days of work during the year received smaller raises than those who missed fewer days. Number of days missed explained 49% of the variation in salary increases. What is the numerical value of the correlation between number of days missed and salary increase?

2.75

.

Question 2.76

2.76 Always plot your data!

Four sets of data prepared by the statistician Frank Anscombe illustrate the dangers of calculating without first plotting the data.11

ansdata

  1. Without making scatterplots, find the correlation and the least-squares regression line for all four data sets. What do you notice? Use the regression line to predict for .
  2. Make a scatterplot for each of the data sets, and add the regression line to each plot.
  3. In which of the four cases would you be willing to use the regression line to describe the dependence of on ? Explain your answer in each case.