91Ó°ÊÓ

Explain the meaning of the words simple and linear as used in simple linear regression.

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. \(y=400-4 x\)

A researcher took a sample of 10 years and found the following relationship between \(x\) and \(y\), where \(x\) is the number of major natural calamities (such as tornadoes, hurricanes, earthquakes, floods, etc.) that occurred during a year and \(y\) represents the average annual total profits (in millions of dollars) of a sample of insurance companies in the United States. $$ \hat{y}=342.6-2.10 x a. A randomly selected year had 24 major calamities. What are the expected average profits of U.S. insurance companies for that year? b. Suppose the number of major calamities was the same for each of 3 years. Do you expect the average profits for all U.S. insurance companies to be the same for each of these 3 years? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact? $$

The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 different varieties of cereal. $$ \begin{array}{l|rrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 \\ \hline \text { Sugar (grams) } & 2 & 7 & 14 & 20 & 3 & 13 & \\ \hline \text { Calories } & 100 & 120 & 190 & 190 & 110 & 120 & \\ \hline \end{array} $$ 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 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 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 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 gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the National League baseball teams. $$ \begin{array}{lcc} \hline \text { Team } & \begin{array}{c} \text { Total Payroll } \\ \text { (millions of dollars) } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Games Won } \end{array} \\ \hline \text { Arizona Diamondbacks } & 92 & 49 \\ \text { Atlanta Braves } & 98 & 41 \\ \text { Chicago Cubs } & 119 & 60 \\ \text { Cincinnati Reds } & 117 & 40 \\ \text { Colorado Rockies } & 102 & 42 \\ \text { Los Angeles Dodgers } & 273 & 57 \\ \text { Miami Marlins } & 68 & 44 \\ \text { Milwaukee Brewers } & 105 & 42 \\ \text { New York Mets } & 101 & 56 \\ \text { Philadelphia Phillies } & 136 & 39 \\ \text { Pittsburgh Pirates } & 88 & 61 \\ \text { San Diego Padres } & 101 & 46 \\ \text { San Francisco Giants } & 173 & 52 \\ \text { St. Louis Cardinals } & 121 & 62 \\ \text { Washington Nationals } & 165 & 51 \\ \hline \end{array} $$ a. Find the least squares regression line with total payroll as the independent variable and percentage of games won as the dependent variable. b. Is the equation of the regression line obtained in part a the population regression line? Why or why not? Do the values of the \(y\) -intercept and the slope of the regression line give \(A\) and \(B\) or \(a\) and \(b ?\) c. Give a brief interpretation of the values of the \(y\) -intercept and the slope obtained in part a. d. Predict the percentage of games won by a team with a total payroll of \(\$ 150\) million.

The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 different varieties of cereal. Here calories is the dependent variable. $$ \begin{array}{l|rrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 \\ \hline \text { Sugar (grams) } & 2 & 7 & 14 & 20 & 3 & 13 & \\ \hline \text { Calories } & 100 & 120 & 190 & 190 & 110 & 120 & \\ \hline \end{array} $$ a. Determine the standard deviation of errors. b. Find the coefficient of determination and give a brief interpretation of it.

While browsing through the magazine rack at a bookstore, a statistician decides to examine the relationship between the price of a magazine and the percentage of the magazine space that contains advertisements. The data collected for eight magazines are given in the following table. Here price is the dependent variable. $$ \begin{array}{l|rrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 \\ \hline \text { Price (\$) } & 5.50 & 6.95 & 4.95 & 5.75 \\ \hline \text { Percentage containing ads } & 70 & 28 & 65 & 32 \\ \hline \text { Price (\$) } & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Find the standard deviation of errors. b. Compute the coefficient of determination. What percentage of the variation in price is explained by the least squares regression of price on the percentage of magazine space containing ads? What percentage of this variation is not explained?

What does a linear correlation coefficient tell about the relationship between two variables? Within what range can a correlation coefficient assume a value?