12. Regression

In linear regression, the least squares method minimizes which of the following quantities?

In a regression analysis using the least squares method, what is the primary objective when fitting a line to the data?

[DATA] Crickets make a chirping noise by sliding their wings rapidly over each other. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The following data list the temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket.

a. What is the most likely explanatory variable in these data? Explain your reasoning.

Correlation and Slope What is the relationship between the linear correlation coefficient r and the slope b1 of a regression line?

Regression and Predictions

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1.

Find the regression equation, letting the first variable be the predictor (x) variable.

Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5.

Powerball Jackpots and Tickets Sold Listed below are the same data from Table 10-1 in the Chapter Problem, but an additional pair of values has been added from actual Powerball results. (Jackpot amounts are in millions of dollars, ticket sales are in millions.) Find the best predicted number of tickets sold when the jackpot was actually 345 million dollars. How does the result compare to the value of 55 million tickets that were actually sold?

[DATA] Home Runs (Refer to Problem 31, Section 4.2) The following data represent the speed at which a ball was hit (in miles per hour) and the distance it traveled (in feet) for a random sample of home runs in a Major League baseball game in 2018.

a. Treating speed at which the ball was hit as the explanatory variable and distance the ball traveled as the response variable, determine estimates for β0 and β1.

The scatterplot below shows a set of data and its least-squares regression line. Based on the graph, which of the following is most likely the equation of the regression line?

Intervals constructed about the predicted value of y, at a given level of x, that are used to measure the accuracy of a single individual’s prediction are called___________intervals for a(n)_________response.

2. Two variables have a positive linear correlation. Is the slope of the regression line for the variables positive or negative?

CEO Performance Explain why it does not make sense to find a least-squares regression line for the CEO Performance data from Problem 33 in Section 4.1.  

[DATA] Bear Markets (Refer to Problem 34, Section 4.1) A bear market is a market condition in which the price of the security falls. A bear market in the stock market is defined as a condition in which the market declines by 20% or more over the course of at least two months. The following data represent the number of months and percentage change in the S&P500 (a group of 500 stocks).

b. Assuming the residuals are normally distributed, test whether a linear relation exists between the number of months of a bear market and percent change at the alpha = 0.05 level of significance.

"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.

5. Black Cherry Tree Volume The volume (in cubic feet) of a black cherry tree can be modeled by the equation

y =- 52.2+0.3x_1 +4.5x_2

where x_1 is the tree's height (in feet) and x_2 is the tree's diameter (in inches). (Source: Journal of the Royal Statistical Society)

a. x_1 = 70, x_2 = 8.6

b. x_1 = 65, x_2 = 11.0

c. x_1 = 83, x_2 = 17.6

d. x_1 = 87, x_2 = 19.6"

[DATA] Bear Markets (Refer to Problem 34, Section 4.1) A bear market is a market condition in which the price of the security falls. A bear market in the stock market is defined as a condition in which the market declines by 20% or more over the course of at least two months. The following data represent the number of months and percentage change in the S&P500 (a group of 500 stocks).

d. Based on your results to parts (b) and (c), would you recommend using the least-squares regression line to predict the percent change in the S&P500 during a bear market? Why?

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

25. Mean Wage Construct a 99% prediction interval for the mean annual wage in Exercise 15 when the percentage of employment in STEM occupations is 13% in the industry."

[DATA] The following data represent the height (inches) of boys between the ages of 2 and 10 years.

d. Assuming the residuals are normally distributed, construct a 95% confidence interval for the slope of the true least-squares regression line.