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Chapter 13: Problem 5

Explain the difference between a simple and a multiple regression model.

Short Answer

Expert verified
A simple regression model predicts the relationship between two variables using a straight line, and is defined by the equation \(Y = a + bX\). A multiple regression model predicts the relationship between one dependent variable and two or more independent variables using a plane or a hyperplane, defined by the equation \(Y = a + b1X1 + b2X2 + ... + bnXn\). The advantage of multiple regression is that it provides a more comprehensive picture when there are numerous predictors.

Step by step solution

01

Define Simple Regression

Simple regression (or simple linear regression) is a statistical method that enables you to summarize and study relationships between two continuous variables: one dependent variable (Y) and one independent variable (X). The model predicts the relationship using a straight line (also known as regression line), defined by the equation \(Y = a + bX\), where 'a' is the y-intercept, 'b' is the slope of the line, Y is the dependent variable and X is the independent variable.
02

Define Multiple Regression

Multiple regression (or multiple linear regression) extends to include more than one independent variable, meaning that it is a statistical procedure that uses two or more predictor variables to forecast the outcome. The model predicts the relationship using a plane or a hyperplane, which is defined by the equation \(Y = a + b1X1 + b2X2 + ... + bnXn\), where 'a' is the y-intercept, 'bi' are the coefficients of the independent variables X1, X2, ..., Xn, Y is the dependent variable.
03

Compare Simple and Multiple Regression

In simple linear regression, we make predictions based on a single predictor, while in multiple linear regression, the prediction is based on more than one predictor. The latter provides a more complete picture when there are multiple predictors due to the additional information about the relationship between these variables and the response.

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Most popular questions from this chapter

The following table lists the midterm and final exam scores for seven students in a statistics class. $$ \begin{array}{l|lllllll} \hline \text { Midterm score } & 79 & 95 & 81 & 66 & 87 & 94 & 59 \\ \hline \text { Final exam score } & 85 & 97 & 78 & 76 & 94 & 84 & 67 \\ \hline \end{array} $$ a. Do you expect the midterm and final exam scores to be positively or negatively related? b. Plot a scatter diagram. By looking at the scatter diagram, do you expect the correlation coefficient between these two variables to be close to zero, 1 , or \(-1\) ? c. Find the correlation coefficient. Is the value of \(r\) consistent with what you expected in parts a and \(\mathrm{b}\) ? d. Using the \(1 \%\) significance level, test whether the linear correlation coefficient is positive.

Suppose that you work part-time at a bowling alley that is open daily from noon to midnight. Although business is usually slow from noon to 6 P.M., the owner has noticed that it is better on hotter days during the summer, perhaps because the premises are comfortably air-conditioned. The owner shows you some data that she gathered last summer. This data set includes the maximum temperature and the number of lines bowled between noon and 6 P.M. for each of 20 days. (The maximum temperatures ranged from \(77^{\circ} \mathrm{F}\) to \(95^{\circ} \mathrm{F}\) during this period.) The owner would like to know if she can estimate tomorrow's business from noon to 6 P.M. by looking at tomorrow's weather forecast. She asks you to analyze the data. Let \(x\) be the maximum temperature for a day and \(y\) the number of lines bowled between noon and 6 P.M. on that day. The computer output based on the data for 20 days provided the following results: \(\hat{y}=-432+7.7 x, \quad s_{e}=28.17, \quad \mathrm{SS}_{x x}=607, \quad\) and \(\quad \bar{x}=87.5\) Assume that the weather forecasts are reasonably accurate. a. Does the maximum temperature seem to be a useful predictor of bowling activity between noon and 6 P.M.? Use an appropriate statistical procedure based on the information given. Use \(\alpha=.05\) b. The owner wants to know how many lines of bowling she can expect, on average, for days with a maximum temperature of \(90^{\circ}\). Answer using a \(95 \%\) confidence level. c. The owner has seen tomorrow's weather forecast, which predicts a high of \(90^{\circ} \mathrm{F}\). About how many lines of bowling can she expect? Answer using a \(95 \%\) confidence level. d. Give a brief commonsense explanation to the owner for the difference in the interval estimates of parts \(\mathrm{b}\) and \(\mathrm{c}\). e. The owner asks you how many lines of bowling she could expect if the high temperature were \(100^{\circ} \mathrm{F}\). Give a point estimate, together with an appropriate warning to the owner.

Explain the meaning of independent and dependent variables for a regression model.

Two variables \(x\) and \(y\) have a negative linear relationship. Explain what happens to the value of \(y\) when \(x\) increases.

Will you expect a positive, zero, or negative linear correlation between the two variables for each of the following examples? a. SAT scores and GPAs of students b. Stress level and blood pressure of individuals c. Amount of fertilizer used and yield of corn per acre d. Ages and prices of houses e. Heights of husbands and incomes of their wives
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