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Chapter 12: Problem 23

Multiple choice: Select the best answer for Exercises 21 to 26. The slope \(\beta\) of the population regression line describes (a) the exact increase in the selling price of an individual unit when its appraised value increases by \(1000. (b) the average increase in the appraised value in a population of units when selling price increases by \)1000. (c) the average increase in selling price in a population of units when appraised value increases by \(1000. (d) the average selling price in a population of units when a unit鈥檚 appraised value is 0. (e) the average increase in appraised value in a sample of 16 units when selling price increases by \)1000.

Short Answer

Expert verified
(c) The average increase in selling price when appraised value increases by 1000.

Step by step solution

01

Understanding the Slope of a Regression Line

The slope \( \beta \) in a regression equation \( y = \beta x + \alpha \) represents the change in the dependent variable (\( y \)) for each one-unit change in the independent variable (\( x \)). It indicates the average change in \( y \) when \( x \) increases by one unit.
02

Identifying Variables

In the context of this problem, the dependent variable is the selling price and the independent variable is the appraised value. Thus, \( \beta \) shows the average change in selling price for each unit increase in appraised value.
03

Applying the Given Increase

We are interested in the change when the appraised value increases by 1000 units. The slope \( \beta \) directly provides this average increase in the selling price.
04

Choosing the Best Answer

Based on our understanding, the best answer is (c): "The average increase in selling price in a population of units when appraised value increases by 1000." This directly matches what the slope of the regression line represents.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Interpretation
The slope of a regression line is a crucial part of understanding how changes in one variable can affect another. It tells us about the relationship between the dependent and independent variables.
Basically, the slope \( \beta \) represents how much we expect the dependent variable to change when the independent variable increases by one unit.
Think of it as the rate of change.
  • If the slope is positive, an increase in the independent variable results in an increase in the dependent variable.
  • If it's negative, an increase in the independent variable results in a decrease in the dependent variable.
So, when interpreting the slope, we are looking at the average change in the dependent variable for every unit change in the independent variable.
It helps to predict and understand how the variables are interacting.
Dependent and Independent Variables
In regression analysis, we deal with two main types of variables: dependent and independent.
The dependent variable, often denoted by \( y \), is the outcome or the variable we are trying to predict or explain.
The independent variable, sometimes denoted by \( x \), is the predictor or the variable we use to explain changes in the dependent variable.
  • In our example, the selling price is the dependent variable. It's what we're interested in understanding or predicting.
  • The appraised value is the independent variable. It's what we manipulate to see how it affects the selling price.
The distinction between these variables is essential because the regression analysis is built on the idea of quantifying this relationship.
Regression Equation
A regression equation is the mathematical way to describe the relationship between the dependent and independent variables.
The general form of the equation is \( y = \beta x + \alpha \), where:
  • \( y \) is the dependent variable we want to predict.
  • \( x \) is the independent variable we use for prediction.
  • \( \beta \) is the slope, telling us how much \( y \) changes for a one-unit change in \( x \).
  • \( \alpha \) (or the intercept) is the value of \( y \) when \( x \) is zero, setting the starting point on the graph.
The regression equation allows us to make predictions and understand the average impact of changes in the independent variable on the dependent variable.
It's a powerful tool that helps in various fields like economics, biology, and social sciences, to interpret and forecast potential outcomes.

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Most popular questions from this chapter

Exercises 51 and 52 refer to the following setting. About 1100 high school teachers attended a weeklong summer institute for teaching AP classes. After hearing about the survey in Exercise 50, the teachers in the AP Statistics class wondered whether the results of the tattoo survey would be similar for teachers. They designed a survey to find out. The class opted for a sample size of 100 teachers. One of the questions on the survey was Do you have any tattoos on your body? (Circle one) YES NO Tattoos (4.1) One of the first decisions the class had to make was what kind of sampling method to use. (a) They knew that a simple random sample was the 鈥減referred鈥� method. With 1100 teachers in 40 different sessions, the class decided not to use an SRS. Give at least two reasons why you think they made this decision. (b) The AP Statistics class believed that there might be systematic differences in the proportions of teachers who had tattoos based on the subject areas that they taught. What sampling method would you recommend to account for this possibility? Explain a statistical advantage of this method over an SRS.

Multiple Choice: Select the best answer for Exercises 45 to 48. Suppose that the relationship between a response variable y and an explanatory variable x is modeled by\(y=2.7(0.316)^{x}\). Which of the following scatterplots would approximately follow a straight line? (a) A plot of y against x (b) A plot of y against log x (c) A plot of log y against x (d) A plot of log y against log x (e) None of (a) through (d)

Exercises 46 to 48 refer to the following setting. Some high school physics students dropped a ball and measured the distance fallen (in centimeters) at various times (in seconds) after its release. If you have studied physics, then you probably know that the theoretical relationship between the variables is distance \(=490(\text { time })^{2}\) . A scatterplot of the students data showed a clear curved pattern. Which of the following single transformations should linearize the relationship? I. time^ \(^{2}\) II. distance \(^{2}\) III. \(\sqrt{\text { distance }}\) (a) I only (c) III only (e) I and III only (b) II only (d) I and II only

Beavers and beetles Do beavers benefit beetles? Researchers laid out 23 circular plots, each four meters in diameter, at random in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth, so that beetles prefer them. If so, more stumps should produce more beetle larvae.\(^{7}\) Minitab output for a regression analysis on these data is shown below. Construct and interpret a 99% confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met. Regression Analysis: Beetle larvae versus Stumps $$ \begin{array}{lcccc} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & -1.286 & 2.853 & -0.45 & 0.657 \\ \text { Stumps } & 11.894 & 1.136 & 10.47 & 0.000 \\ \mathrm{~S}=6.41939 & \mathrm{R}-\mathrm{Sq} & =83.98\% &\mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =83.18\% \end{array} $$

Multiple choice: Select the best answer for Exercises 21 to 26. What is the correlation between selling price and appraised value? (a) 0.1126 (c) -0.861 (e) -0.928 (b) 0.861 (d) 0.928
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