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Chapter 3: Q 51. (page 204)

Movie candy Is there a relationship between the amount of sugar (in grams) and the number of calories in movie-theater candy? Here are the data from a sample of 12 types of candy:

a. Sketch a scatterplot of the data using sugar as the explanatory variable.

b. Use technology to calculate the equation of the least-squares regression line for predicting the number of calories based on the amount of sugar. Add the line to the scatterplot from part (a).

c. Explain why the line calculated in part (b) is called the “least-squares” regression line.

Short Answer

Expert verified

Part (b) The regression line is: y⏜=2.829x+300.040

Part (c) It's known as a least square regression line.

Part (a)

Step by step solution

01

Part (a) Step 1: Given information

The data set is

02

Part (a) Step 2: Concept

The least-squares regression line reduces the sum of squares of vertical distances between the observed points and the line to zero.

03

Part (a) Step 3: Explanation

The scatter plot is:

04

Part (b) Step 1: Explanation

The output obtained using the Ti-83 calculator is:

Thus, the regression line is: y⏜=2.829x+300.040

The regression line on the scatter plot could be shown as:

05

Part (c) Step 1: Explanation

Because it is a regression line, the regression line generated in the previous section minimizes the errors or residuals. As a result, it's known as a least square regression line.

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Most popular questions from this chapter

Long-term records from the Serengeti National Park in Tanzania show interesting ecological relationships. When wildebeest are more abundant, they graze the grass more heavily, so there are fewer fires and more trees grow. Lions feed more successfully when there are more trees, so the lion population increases. Researchers collected data on one part of this cycle, wildebeest abundance (in thousands of animals) and the percent of the grass area burned in the same year. The results of a least-squares regression on the data are shown here.


a. Is a linear model appropriate for describing the relationship between wildebeest abundance and percent of grass area burned? Explain.

b. Give the equation of the least-squares regression line. Be sure to define any variables you use.

c. Interpret the slope. Does the value of the y-intercept have meaning in this context? If so, interpret the y-intercept. If not, explain why.

d. Interpret the standard deviation of the residuals and r2.

Fruit fly thorax lengths (2.2) Fruit flies are used frequently in genetic research because of their quick reproductive cycle. The length of the thorax (in millimeters) for male fruit flies is approximately Normally distributed with a mean of 0.80 mm and a standard deviation of 0.08 mm.21

a. What proportion of male fruit flies have a thorax length greater than 1 mm?

b. What is the 30th percentile for male fruit fly thorax lengths?

More Olympics Athletes who participate in the shot put, discus throw, and hammer throw tend to have different physical characteristics than other track and field athletes. The scatterplot shown here enhances the scatterplot from Exercise 5 by plotting these athletes with blue squares. How are the relationships between height and weight the same for the two groups of athletes? How are the relationships different?

Late bloomers? Japanese cherry trees tend to blossom early when spring weather is warm and later when spring weather is cool. Here are some data on the average March temperature (in degrees Celsius) and the day in April when the first cherry blossom appeared over a 24-year period:

a. Make a well-labeled scatterplot that’s suitable for predicting when the cherry trees will blossom from the temperature. Which variable did you choose as the explanatory variable? Explain your reasoning.

b. Use technology to calculate the correlation and the equation of the least-squares regression line. Interpret the slope and y-intercept of the line in this setting.

c. Suppose that the average March temperature this year was 8.2°C. Would you be willing to use the equation in part (b) to predict the date of the first blossom? Explain your reasoning.

d. Calculate and interpret the residual for the year when the average March temperature was 4.5°C.

e. Use technology to help construct a residual plot. Describe what you see.

The scatterplot shows the lean body mass and metabolic rate for a sample of = adults. For each person, the lean body mass is the subject’s total weight in kilograms less any weight due to fat. The metabolic rate is the number of calories burned in a 24-hour period.

Because a person with no lean body mass should burn no calories, it makes sense to model the relationship with a direct variation function in the form y = kx. Models were tried using different values of k (k = 25, k = 26, etc.) and the sum of squared residuals (SSR) was calculated for each value of k. Here is a scatterplot showing the relationship between SSR and k:

According to the scatterplot, what is the ideal value of k to use for predicting metabolic rate?

a. 24

b. 25

c. 29

d. 31

e. 36
See all solutions

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