12. Regression

[DATA] Concrete As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

c. Determine sb_1.

[DATA] Graduation Rates PayScale reports statistics on colleges and universities. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 11_3_24 using the file format of your choice for the version of the text you are using. The data contain the four-year cost and graduation rate for over 1300 colleges and universities. Do schools that charge more have higher graduation rates? The variable “4 Year Cost” represents the four-year cost of attending the college or university. The variable “Grad Rate” represents the percentage of incoming freshman who graduate within six years.

f. What proportion of the variability in graduation rates is explained by the cost of attending?

Suppose a least-squares regression line is given by ŷ = 4.302x – 3.293. What is the mean value of the response variable if x = 20?

[DATA] Invest in Education Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_3_17. The variable “Cost” represents the four-year cost including tuition, supplies, room and board, the variable “Annual ROI” represents the return on investment for graduates of the school—essentially how much you would earn on the investment of attending the school. The variable “Grad Rate” represents the graduation rate of the school.

c. What is the mean annual ROI for a four-year school whose cost is \$180,000?

[DATA] CEO Performance (Refer to Problem 33 in Section 4.1) The following data represent the total compensation for 12 randomly selected chief executive officers (CEOs) and the company’s stock performance in 2017.

b. Assuming the residuals are normally distributed, test whether a linear relation exists between compensation and stock return at the alpha = 0.05 level of significance

[DATA] CEO Performance (Refer to Problem 33 in Section 4.1) The following data represent the total compensation for 12 randomly selected chief executive officers (CEOs) and the company’s stock performance in 2017.

d. Based on your results to parts (b) and (c), would you recommend using the least-squares regression line to predict the stock return of a company based on the CEO’s compensation? Why?

[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.

"[DATA] An Unhealthy Commute [See Problem 11 in Section 12.3] The following data represent commute times (in minutes) and a score on a well-being survey.

b. Determine the slope of the least-squares regression line between commute time and well-being score."

What do the y-coordinates on the least-squares regression line represent?

In the least-squares regression model, yi = β1xi + β0 + εi, εi is a random error term with mean ______ and standard deviation σεi = ______.

[DATA] Tar and Nicotine Every year the Federal Trade Commission (FTC) must report tar and nicotine levels in cigarettes to Congress. Tar and nicotine levels of over 1200 brands of cigarettes are given to Congress and a random sample of those appear in the following table:

c. Determine sb_1.

In Problems 5–10, use the results of Problems 7–12, respectively, from Section 4.2 to answer the following questions:

a. What are the estimates of β₀ and β₁?

The output shown was obtained from Minitab.

a. The least-squares regression equation is y^ = 1.3962x + 12.396. What is the predicted value of y at x = 10?

The output shown was obtained from Minitab.

c. The standard error, se, is 2.167. What is an estimate of the standard deviation of y at x=10?

[DATA] Height versus Head Circumference [See Problem 13 in Section 12.3] A pediatrician wants to determine the relation that may exist between a child’s height and head circumference. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the following data:

d. State your conclusion to the hypotheses from part (b).