Choosing the Best Model Answers

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.

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.

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?

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 .

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?

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.

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?

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.

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.

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 .

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.

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 .

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.

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.

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?

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.

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?

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.

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?

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.