91Ó°ÊÓ

Explain the difference between exact and nonexact relationships between two variables. Give one example of each.

Explain the difference between linear and nonlinear relationships between two variables.

Why is the random error term included in a regression model?

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\). a. \(y=100+5 x\) b. \(y=400-4 x\)

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\) a. \(y=-60+8 x\) b. \(y=300-6 x\)

A population data set produced the following information. $$ N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080, \quad \Sigma x^{2}=485,870 $$ Find the population regression line.

A car rental company charges $$\$ 50$$ a day and 20 cents per mile for renting a car. Let \(y\) be the total rental charges (in dollars) for a car for one day and \(x\) be the miles driven. The equation for the relationship between \(x\) and \(y\) is $$ y=50+.20 x $$ a. How much will a person pay who rents a car for one day and drives it 100 miles? b. Suppose each of 20 persons rents a car from this agency for one day and drives it 100 miles. Will each of them pay the same amount for renting a car for a day or do you expect each person to pay a different amount? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?

An auto manufacturing company wanted to investigate how the price of one of its car models depreciates with age. The research department at the company took a sample of eight cars of this model and collected the following information on the ages (in years) and prices (in hundreds of dollars) of these cars. $$ \begin{array}{l|rrrrrrrr} \hline \text { Age } & 8 & 3 & 6 & 9 & 2 & 5 & 6 & 3 \\ \hline \text { Price } & 45 & 210 & 100 & 33 & 267 & 134 & 109 & 235 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between ages and prices of cars? b. Find the regression line with price as a dependent variable and age as an independent variable. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\mathrm{b}\). d. Plot the regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the regression line. e. Predict the price of a 7 -year-old car of this model. \(\mathbf{f}\). Estimate the price of an 18 -year-old car of this model. Comment on this finding.

The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 varieties of Kellogg's cereal. $$ \begin{array}{l|rrrrrrrrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 & 2 & 7 & 14 & 20 & 3 & 13 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 & 100 & 120 & 190 & 190 & 110 & 120 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the amount of sugar and the number of calories per serving? b. Find the predictive regression equation of the number of calories on the amount of sugar. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the predictive regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Calculate the predicted calorie count for a cereal with 16 grams of sugar per serving. f. Estimate the calorie count for a cereal with 52 grams of sugar per serving. Comment on this finding.

The following table contains information on the amount of time that each of 12 students spends each day (on average) on social networks (Facebook, Twitter, etc.) and the Internet for social or entertainment purposes and his or her grade point average (GPA). $$ \begin{array}{l|rrrrrrrrrrrr} \hline \text { Time (hours per day) } & 4.4 & 6.2 & 4.2 & 1.6 & 4.7 & 5.4 & 1.3 & 2.1 & 6.1 & 3.3 & 4.4 & 3.5 \\ \hline \text { GPA } & 3.22 & 2.21 & 3.13 & 3.69 & 2.7 & 2.2 & 3.69 & 3.25 & 2.66 & 2.89 & 2.71 & 3.36 \\ \hline \end{array} $$a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between grade point average and time spent on social networks and the Internet? b. Find the predictive regression line of GPA on time. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the predictive regression line on the scatter diagram of part a, and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Calculate the predicted GPA for a college student who spends \(3.8\) hours per day on social networks and the Internet for social or entertainment purposes. f. Calculate the predicted GPA for a college student who spends 16 hours per day on social networks and the Internet for social or entertainment purposes. Comment on this finding.