12. Regression

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

What is the simple least-squares regression model? What are the requirements to perform inference on a simple least-squares regression line? How do we verify that these requirements are met?

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

29. New Vehicle Sales Construct a 95% prediction interval for new vehicle sales for General Motors in Exercise 19 when the number of new vehicles sold by Ford is 2028 thousand."

Given a linear regression equation of the form $y=a+bx$ with $b=3$ and $a=-6$, what is the predicted value of $y$ when $x=4$?

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

21. Proceeds Construct a 95% prediction interval for the proceeds from initial public offerings in Exercise 11 when the number of offerings is 200."

[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:

f. What is the estimated mean 28-day strength of this concrete if the 7-day strength is 3000 psi?

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

27. Natural Gas Construct a 95% prediction interval for the export of natural gas from the United States in Exercise 17 when the marketed production of natural gas in the United States is 31 trillion cubic feet."

Given the following data points: $(1,2)$, $(2,3)$, $(3,5)$, use the least squares method to find the equation of the regression line in the form $y=a+bx$. Which of the following is the correct regression equation?

Bear Markets Explain why it does not make sense to find a least-squares regression line for the Bear Market data from Problem 34 in Section 4.1.  

Which of the following represents the equation of the least-squares regression line for predicting $y$ from $x$?

[DATA] CEO Performance [See Problem 19 in Section 12.3] The following data represent the total compensation for 12 randomly selected chief executive officers (CEOs) and the company’s stock performance in 2017.

b. Suppose a researcher wanted to determine if there is a linear relation between compensation and stock return. What would be the null and alternative hypotheses?

Testing Hypotheses About Regression Coefficients If the coefficient has a nonzero value, then it is helpful in predicting the value of the response variable. If it is not helpful in predicting the value of the response variable and can be eliminated from the regression equation. To test the claim that use the test statistic Critical values or P-values can be found using the t distribution with degrees of freedom, where k is the number of predictor variables and n is the number of observations in the sample. The standard error is often provided by software. For example, see the accompanying StatCrunch display for Example 1, which shows that (found in the column with the heading of “Std. Err.” and the row corresponding to the first predictor variable of height). Use the sample data in Data Set 1 “Body Data” and the StatCrunch display to test the claim that Also test the claim that What do the results imply about the regression equation?

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3. Explain how to predict y-values using the equation of a regression line.

Using the sample data from Problem 6 in Section 12.3,

a. Predict the mean value of y if x=8.