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Choosing the Best Model Answers

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Data were recorded for the temperature of a cup of coffee over a 30-minute period. A regression analysis was completed and the computer output is given.

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Data were recorded for the temperature of a cup of coffee over a 30 minute period. It is known that the temperature of hot coffee will cool to room temperature following an exponential model.Which of the following would linearize the data for temperature and minutes?

A
Minutes, Temperature
B
Minutes, ln(Temperature)
C
ln(Minutes), Temperature
D
ln(Minutes), ln(Temperature)
3

Data were recorded for the temperature, in degrees Celsius, of a cup of coffee over a 30-minute period. Given the regression equation, , what is the predicted temperature after 5 minutes?

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59.44 oC
B
64.26 oC
C
65.17 oC
D
66.69 oC
4

Water is being poured into a large, cone-shaped cistern. The volume of water, measured in cm3, is reported at different time intervals, measured in seconds and is displayed in scatterplot A. Two transformations of the data are shown in the second and third graphs. Scatterplot B displays the relationship between time and the log of volume. Scatterplot C displays the relationship between the log of time and the log of volume. Which model best represents the relationship between time and volume?

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A
A linear model would best represent the relationship because scatterplot A is fairly linear.
B
An exponential model would best represent the relationship because scatterplot B is fairly linear.
C
A power model would best represent the relationship because scatterplot C is fairly linear.
D
A linear model would best represent the relationship because scatterplot C is fairly linear.
5

Data were recorded for a car’s fuel efficiency, in miles per gallon (mpg), and the corresponding speed, in miles per hour (mph). A regression analysis was completed and the computer output is given.

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When a stone is dropped in a pond, ripples are formed and travel in concentric circles away from where the stone was dropped. The relationship between the time and area of the circles was analyzed and is shown in the computer output.

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Water is being poured into a large, cone-shaped cistern. The volume of water, measured in cm3, is reported at different time intervals, measured in seconds. A regression analysis was completed and is displayed in the computer output.

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The regression equation for water being poured into a large, cone-shaped cistern is . What is the predicted volume for a time of 12 seconds?

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A
1.810 cm3
B
6.110 cm3
C
34.589 cm3
D
450.485 cm3
9

The lengths and weights for 15 different-sized bluegill fish were measured and recorded. Scatterplot A displays the relationship between the length, in inches, and the weight, in ounces, for each fish. Two transformations of the data are shown in the scatterplots B and C. Scatterplot B displays the relationship between the length and the natural log of the weight. Scatterplot C displays the relationship between the natural log of the length and the natural log of the weight.Which model best represents the relationship between the bluegills’ lengths and weights?

Question illustration
A
A linear model would best represent the relationship because scatterplot A is fairly linear.
B
An exponential model would best represent the relationship because scatterplot B is fairly linear.
C
A linear model would best represent the relationship because scatterplot C is fairly linear.
D
A power model would best represent the relationship because scatterplot C is fairly linear.
10

The lengths, in inches, and corresponding weights, in ounces, for 15 different-sized bluegill fish were measured and recorded. Given the regression equation, , what is the expected weight for a bluegill fish that is 11 inches long?

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A
19.47 ounces
B
31.19 ounces
C
32.56 ounces
D
37.84 ounces
11

Online entertainment streaming services have gained in popularity in recent years as an alternative to traditional television. One such company has seen steady growth in each period of 3 months, called a quarter, over the past 4 years. The scatterplot shows the relationship between the number of quarters passed since January 2014 and the number of members to the streaming service. A least-squares equation that summarizes this relationship is , with a value for r of 0.991.

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A
A linear model is appropriate because the value of r is close to 1.
B
A linear model is appropriate because the residual plot is curved.
C
A linear model is not appropriate because the residual plot shows a clear pattern.
D
A linear model is not appropriate because the scatterplot relating time and subscribers shows a clear pattern.
12

Many Texas cities have experienced substantial growth in population over the last 20 years. The growth of Houston, Texas, since 2015 is shown in the scatterplot. A least-squares equation that relates the number of years since 2015 to the population of that year (in millions) is given by .

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A
A linear model is suitable because a pattern is evident in the residual plot.
B
A linear model is not suitable because the residuals do not cluster around zero.
C
A linear model is not suitable because the residual plot shows a curved pattern.
D
A linear model is suitable because the residuals seem to be randomly scattered.
13

As an object travels away from a light source, the intensity of the light on the object diminishes. To measure the influence of distance on light intensity, a student uses a light meter to record intensity, in lumens, from a source at various distances. The results, which compare distance in centimeters to the recorded light intensity, are shown in the scatterplot. To develop a linear model, the student next took the log of each distance and the log of each intensity and used computer software to find a least-square equation, shown in the computer output.

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A
0.0876, because 0.8561 − 1.4966(log 19) = −1.058, and 10−1.058=0.0876 lumens.
B
0.3472, because 0.8561 − 1.4966(log 19) = −1.058, and e−1.058=0.3472 lumens.
C
0.3964, because 0.8561(log 19) − 1.4966 = −0.4018, and 10−0.4018=0.3964 lumens.
D
0.6691, because 0.8561(log 19) − 1.4966 = −0.4018, and e−0.4018=0.6691 lumens.
14

A small technology company started offering shares of stock to investors in 1987. At that time, the price of one share of stock was $0.39. Since then, the company has experienced rapid growth. Twenty-two years later, the price of a single share of stock has risen to over $110. One scatterplot shown summarizes the number of years since the initial stock offering in 1987 and the price of the stock, while the other compares years since 1987 to the log of stock price.

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A
A linear model is appropriate because the scatterplot of the transformed data is roughly linear.
B
An exponential model is appropriate because the scatterplot of time and log price is increasing.
C
An exponential model could be appropriate because the scatterplot of time versus log price is roughly linear.
D
A power model is appropriate because the scatterplot of time and price shows a curved relationship.
15

A forest management agency collects measurements from beech trees in a public park. The age of many of the trees is known, and the basal area—the area of land occupied by the trunks of the trees—is also measured. A least-squares equation that compares the basal area to the age of each tree is given by .

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Because the residuals are centered about zero, a linear model is appropriate.
B
Because the residual plot does not show a clear pattern, a linear model is appropriate.
C
Because the residual plot shows a random a scatter of points, a linear model is not appropriate.
D
Because the residual plot does not appear linear, it is likely that tree age and basal area do not have a linear relationship.
16

Many new cars provide detailed information about engine performance on the dashboard. One such feature allows drivers to observe current fuel efficiency, recorded in miles per gallon, as they drive. A consumer takes a long trip driving at different speeds, while a passenger records both driving speed in miles per hour and fuel efficiency for a number of selected points along the trip. A least-squares equation that relates speed to fuel efficiency is given by .

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A
A linear model is appropriate because the residual plot is clearly curved.
B
A linear model is not appropriate because the residual plot shows a clear pattern.
C
A linear model is appropriate because the residuals are decreasing at higher car speeds.
D
A linear model is not appropriate because there are more negative residuals than positive residuals.
17

An official for a national dog show studies the characteristics of one breed of dog, the Dandie Dinmont Terrier. Two common measurements are the height and weight of the dog, and the official would like to develop a model that would be helpful in predicting weight based on a given height. The official first makes a scatterplot that relates height and weight, then another that compares the logs of each measurement.

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A
A linear model is appropriate because the graph of the transformed data is roughly linear.
B
A power model is appropriate because the scatterplot of height versus weight appears curved.
C
A power model could be appropriate because the scatterplot of log height versus log weight is roughly linear. The next step is to look at the residual plot.
D
An exponential model is appropriate because the scatterplot of log height versus log weight has a stronger linear relationship than the scatterplot of the non-transformed data.
18

A punter for a football team is trying to determine the optimal angle for striking the football off his foot—this is called the launch angle. Using video, his coach records a number of punts kicked using different launch angles and the height in feet for each punt. Using computer software, the coach achieves a linear model by taking the log of each launch angle and plotting against the log of each height.

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A
A linear model is appropriate because the residual plot shows a random scatter.
B
A logarithmic model is appropriate because logarithms were used to transform the data sets.
C
A power model is appropriate because the scatterplot of log angle and log height is roughly linear.
D
An exponential model is appropriate because the scatterplot of log angle and log height is roughly linear.
19

The daily print publication of newspapers has declined in many large cities over the past 25 years. The Philadelphia Inquirer weekday print circulation was at its highest in 2001, with an average daily circulation of 226,000 copies, and has decreased every year since then. To develop a linear model for estimating yearly circulation, the log of the daily circulation in thousands was taken.

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A
A linear model is appropriate because the residual plot does not show a clear pattern.
B
A power model is appropriate because the scatterplot of years and the log of circulation is roughly linear.
C
An exponential model is appropriate because the scatterplot of years and the log of circulation is roughly linear and the residual plot shows no distinct pattern.
D
Both an exponential and a power model would be appropriate because the log of circulation was used to develop the model.
20

One method used to measure the speed of supercomputers is the number of floating-point mathematical operations the computer can perform in one second. This is often referred to by the acronym FLOPS. For many years since 1992, the number of FLOPS performed by the largest supercomputer available that year was recorded, and the natural log of each value of the response variable taken.

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A
13.3 billion FLOPS because ŷ = 06498(15)+305246 = 13.2716
B
17,334 billion FLOPS because and e9.7604 = 17,334.
Option B
C
580,473 billion FLOPS because and e13.2716= 580,473.
Option C
D
5,759,701,813 billion FLOPS because and 109.7604 = 5,759,701,813.
Option D

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