Question 1

In simple linear regression, the least squares line fit to a sample of data will maximize the number of data points that will fall along that line.

Accepted Answer

The least squares line minimizes the sum of the squares of the vertical distances (residuals) of the points from the line, not the number of points that fall exactly on the line.

Question 2

A procedure used for finding the equation of a straight line that provides the best fit by minimizing the sum of the squared vertical distances of points from the line is called the least squares method.

Accepted Answer

The least squares method is indeed used to find the equation of a straight line that best fits a set of data points by minimizing the sum of the squares of the vertical distances of the points from the line.

Question 3

In simple linear regression, the slope and intercept values for the least squares line fit to a sample of data points serve as point estimates of the slope and intercept terms of the least squares line that would be fit to the population of data points.

Accepted Answer

In simple linear regression, the slope and intercept obtained from a sample are indeed used as estimates for the slope and intercept that would be obtained if we were able to fit the regression line to the entire population. These sample estimates aim to infer the population parameters.

Question 4

In multiple regression, a single independent variable is linked to a set of two or more dependent variables.

Accepted Answer

In multiple regression, a single dependent variable is linked to a set of two or more independent variables.

Question 5

In simple linear regression, there is an additive relationship between SST, SSR, and SSE: that is, SST = SSR + SSE

Accepted Answer

In simple linear regression, the total sum of squares (SST) is equal to the sum of the regression sum of squares (SSR) and the sum of squared errors (SSE), reflecting the partitioning of variance into explained and unexplained parts.

Question 6

In simple linear regression, the r² value measures the percentage of total variation in the sample data that cannot be explained by the x-to-y relationship that has been identified.

Accepted Answer

In simple linear regression, the $R^2$ value measures the percentage of total variation in the dependent variable (y) that can be explained by the independent variable (x) through the linear relationship.

Question 7

The coefficient of determination r², a measure of the goodness of fit of the estimated regression equation, is the ratio of the explained variation in y to the total variation in y, and takes on values between 0 and 1.

Accepted Answer

The coefficient of determination, often denoted as $R^2$, indeed measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s), ranging from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect explanatory power.

Question 8

In a simple linear regression, the least squares line turns out to be y = 120 + 4x.The observed value of y when x = 10 is 182.The residual value for x = 10 must be 22.

Accepted Answer

The residual value is the difference between the observed value and the predicted value. The predicted value for x = 10 is 120 + 4(10) = 160. Therefore, the residual value is 182 - 160 = 22.

Question 9

In simple regression, the usual goal is to identify a cause-and-effect relationship between two variables.

Accepted Answer

The usual goal of simple regression is to model the relationship between two variables, typically predicting one variable based on the other, rather than to establish a cause-and-effect relationship.

Question 10

In residual analysis, if the assumptions about the error term are valid then the plot of the residuals against the corresponding x values should have a funnel shape.

Accepted Answer

The plot of the residuals against the corresponding x values should ideally show no particular pattern, indicating homoscedasticity (constant variance) of errors, rather than a funnel shape which indicates heteroscedasticity (non-constant variance).

Question 11

Residual analysis makes use of a plot of residuals (on the y axis) and the independent variable (on the x axis) to evaluate the assumptions about the error term in regression.

Accepted Answer

Residual analysis involves plotting residuals against the independent variable to assess the assumptions regarding the error terms in a regression model, such as homoscedasticity and independence of errors.

Question 12

In simple linear regression, the least squares line fit to a sample of data points seeks to minimize the maximum distance of points from the line.

Accepted Answer

The least squares line fit actually seeks to minimize the sum of the squared vertical distances (residuals) of the points from the line, not the maximum distance of any point from the line.

Question 13

In simple linear regression, rejecting the null hypothesis in the key hypothesis test regarding the slope of the "population" line means we haven't yet found sufficient sample evidence that there's a useful linear relationship between x and y.

Accepted Answer

Rejecting the null hypothesis in the key hypothesis test regarding the slope of the "population" line actually means we have found sufficient sample evidence to suggest there is a useful linear relationship between x and y.

Question 14

In simple linear regression, the difference between a predicted y value and an observed y value is commonly called the residual or error value.

Accepted Answer

In simple linear regression, the difference between the actual value of y and the predicted value of y is indeed called the residual or error. This measures how far the data points are from the regression line.

Question 15

When the slope of a simple linear regression equation is a negative value, the correlation coefficient can be either positive or negative.

Accepted Answer

The correlation coefficient, which ranges from -1 to 1, indicates the direction of the relationship between two variables. If the slope of a simple linear regression equation is negative, it means that as one variable increases, the other decreases, indicating a negative correlation. Therefore, the correlation coefficient must also be negative.

Question 16

In a simple linear regression, the least squares line turns out to be y = 120 + 4x.The sum of the residuals (errors) around this line will be 0.

Accepted Answer

In simple linear regression, the least squares method ensures that the sum of the residuals (the differences between the observed values and the values predicted by the model) is zero. This is a property of the least squares estimation.

Question 17

A procedure used for finding the equation of a straight line that provides the best fit for the relationship between the independent and dependent variables is called the minimum deviations method.

Accepted Answer

The correct term for the procedure used to find the equation of a straight line that provides the best fit for the relationship between the independent and dependent variables is the "least squares method," not the minimum deviations method.

Question 18

In simple linear regression, the r² value is the ratio of SSE/SST.

Accepted Answer

In simple linear regression, the $R^2$ value, not $r²$, is the ratio of SSR (Sum of Squares due to Regression) to SST (Total Sum of Squares), which represents the proportion of variance in the dependent variable that is predictable from the independent variable.

Question 19

The standard error of estimate (s_y.x) in simple linear regression is the square of the Mean Square Error (MSE) which is the sum of squares error (SSE) multiplied by its degrees of freedom n-2.

Accepted Answer

The standard error of the estimate in simple linear regression is the square root of the Mean Square Error (MSE), not the square of MSE. MSE is calculated as the sum of squares error (SSE) divided by its degrees of freedom (n-2), not multiplied.

Question 20

In simple linear regression, the variable that will be predicted is labeled the dependent variable.

Accepted Answer

In simple linear regression, the variable that is being predicted or explained is called the dependent variable, while the variable used to predict the value of the dependent variable is called the independent variable.

Quiz 11: Basic Regression Analysis