Question 1

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line:  = 84 +7X. This implies that if there is no advertising, then the predicted amount of sales (in dollars) is $84,000.

Accepted Answer

The intercept on the least squares line represents the predicted sales when there is no advertising. In this case, the intercept is $84, which means the predicted sales without any advertising is $84,000.

Question 2

In regression analysis, we can often use the standard error of estimate  to judge which of several potential regression equations is the most useful.

Accepted Answer

The standard error of estimate is a measure of the accuracy of the regression equation in predicting the dependent variable. Therefore, a smaller standard error of estimate indicates a more useful regression equation, and can be used to compare several potential equations.

Question 3

When the scatterplot appears as a shapeless swarm of points, this can indicate that there is no relationship between the response variable Y and the explanatory variable X, or at least none worth pursuing.

Accepted Answer

This is a correct statement. A shapeless swarm of points in a scatterplot suggests that there is no clear pattern or relationship between the two variables being compared. It is possible that there may be some other factors influencing the relationship, but it is unlikely that there is a strong, meaningful connection between the two variables.

Question 4

Correlation is measured on a scale from 0 to 1, where 0 indicates no linear relationship between two variables, and 1 indicates a perfect linear relationship.

Accepted Answer

Correlation is measured on a scale from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship between two variables.

Question 5

The least squares line is the line that minimizes the sum of the residuals.

Accepted Answer

The least squares line minimizes the sum of the squares of the residuals, not the sum of the residuals themselves.

Question 6

In a simple linear regression problem, if the percentage of variation explained  is 0.95, this means that 95% of the variation in the explanatory variable X can be explained by regression.

Accepted Answer

The percentage of variation explained (R-squared) refers to the proportion of variation in the response variable Y that can be explained by the regression model, not the explanatory variable X.

Question 7

The residual is defined as the difference between the actual and predicted, or fitted values of the response variable.

Accepted Answer

This is the correct definition of a residual.

Question 8

Correlation is used to determine the strength of the linear relationship between an explanatory variable X and response variable Y.

Accepted Answer

Correlation measures the strength of the linear relationship between two variables.

Question 9

An outlier is an observation that falls outside of the general pattern of the rest of the observations on a scatterplot.

Accepted Answer

An outlier is an observation that is significantly different from other observations in a dataset and falls outside of the general pattern on a scatterplot.

Question 10

A negative relationship between an explanatory variable X and a response variable Y means that as X increases, Y decreases, and vice versa.

Accepted Answer

A negative relationship between two variables means that as one variable increases, the other variable decreases.

Question 11

Scatterplots are used for identifying outliers and quantifying relationships between variables.

Accepted Answer

The answer of Scatterplots are used for identifying outliers and...

Question 12

In simple linear regression, the divisor of the standard error of estimate  is n - 1, simply because there is only one explanatory variable of interest.

Accepted Answer

The answer of In simple linear regression, the divisor of...

Question 13

A regression analysis between sales (in $1000) and advertising (in $) resulted in the following least squares line:  = 32 + 8X. This implies that an increase of $1 in advertising is expected to result in an increase of $40 in sales.

Accepted Answer

An increase of $1 in advertising is expected to result in an increase of $8 in sales, not $40. The statement in the question is incorrect.

Question 14

In a simple regression analysis, if the standard error of estimate  = 15 and the number of observations n = 10, then the sum of the residuals squared must be 120.

Accepted Answer

The sum of the residuals squared is equal to the product of the standard error of estimate and the square root of the degrees of freedom, which is n-2 in simple regression. Therefore, the sum of the residuals squared is equal to 15 * sqrt(10-2) = 47.43, not 120.

Question 15

The multiple R for a regression is the correlation between the observed Y values and the fitted Y values.
.

Accepted Answer

The multiple R for a regression is the correlation coefficient between the observed Y values and the fitted Y values.

Question 16

A regression analysis between weight (Y in pounds) and height (X in inches) resulted in the following least squares line:  = 140 + 5X. This implies that if the height is increased by 1 inch, the weight is expected to increase on average by 5 pounds.

Accepted Answer

The statement in the question is exactly what the slope of the least squares line (5) represents: the expected increase in weight for every 1 unit increase in height. Therefore, the statement is true.

Question 17

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line:  = 84 +7X. This implies that if advertising is $800, then the predicted amount of sales (in dollars) is $140,000.

Accepted Answer

If advertising is $800 (X=8), then the predicted amount of sales (in dollars) is:
11eb06d9_cc88_a911_8177_3952683857ab_TB1387_12 = 84 + 7(8) = 140
Therefore, the predicted amount of sales in $1000 is $140,000. So, the statement is true.

Question 18

A useful graph in almost any regression analysis is a scatterplot of residuals (on the vertical axis) versus fitted values (on the horizontal axis), where a "good" fit not only has small residuals, but it has residuals scattered randomly around zero with no apparent pattern.

Accepted Answer

This is a commonly used graph to check the assumptions of the regression model, and a "good" fit is indeed characterized by residuals scattered randomly around zero with no clear pattern.

Question 19

In reference to the equation,  , the value 0.10 is the expected change in Y per unit change in  .

Accepted Answer

Based on the equation given, the coefficient of 11eb06d9_cc88_a910_8177_4b975b5ad46a_TB1387_11 is 0.10, which means that a unit increase in 11eb06d9_cc88_a910_8177_4b975b5ad46a_TB1387_11 will result in a 0.10 unit increase in Y. Therefore, the statement is true.

Question 20

The regression line  = 3 + 2X has been fitted to the data points (4, 14), (2, 7), and (1, 4). The sum of the residuals squared will be 8.0.

Accepted Answer

To find the sum of residuals squared, we need to first calculate the predicted values using the regression line, and then subtract them from the actual values to get the residuals. Then we square each residual, and add them up.

Using the regression line 11eb06d9_cc89_1e47_8177_adf4b08f8323_TB1387_11 = 3 + 2X, we get:
- For the point (4, 14):
Predicted value = 3 + 2(4) = 11
Residual = actual - predicted = 14 - 11 = 3
Residual squared = 3^2 = 9

- For the point (2, 7):
Predicted value = 3 + 2(2) = 7
Residual = actual - predicted = 7 - 7 = 0
Residual squared = 0^2 = 0

- For the point (1, 4):
Predicted value = 3 + 2(1) = 5
Residual = actual - predicted = 4 - 5 = -1
Residual squared = (-1)^2 = 1

So the sum of residuals squared is 9 + 0 + 1 = 10, not 8. Therefore, the statement is false and the answer is B.

Quiz 10: Regression Analysis: Estimating Relationships

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line: = 84 +7X. This implies that if there is no advertising, then the predicted amount of sales (in dollars) is $84,000.

In regression analysis, we can often use the standard error of estimate to judge which of several potential regression equations is the most useful.

When the scatterplot appears as a shapeless swarm of points, this can indicate that there is no relationship between the response variable Y and the explanatory variable X, or at least none worth pursuing.

Correlation is measured on a scale from 0 to 1, where 0 indicates no linear relationship between two variables, and 1 indicates a perfect linear relationship.

The least squares line is the line that minimizes the sum of the residuals.

In a simple linear regression problem, if the percentage of variation explained is 0.95, this means that 95% of the variation in the explanatory variable X can be explained by regression.

The residual is defined as the difference between the actual and predicted, or fitted values of the response variable.

Correlation is used to determine the strength of the linear relationship between an explanatory variable X and response variable Y.

An outlier is an observation that falls outside of the general pattern of the rest of the observations on a scatterplot.

A negative relationship between an explanatory variable X and a response variable Y means that as X increases, Y decreases, and vice versa.

Scatterplots are used for identifying outliers and quantifying relationships between variables.

In simple linear regression, the divisor of the standard error of estimate is n - 1, simply because there is only one explanatory variable of interest.

A regression analysis between sales (in $1000) and advertising (in $) resulted in the following least squares line: = 32 + 8X. This implies that an increase of $1 in advertising is expected to result in an increase of $40 in sales.

In a simple regression analysis, if the standard error of estimate = 15 and the number of observations n = 10, then the sum of the residuals squared must be 120.

The multiple R for a regression is the correlation between the observed Y values and the fitted Y values. .

A regression analysis between weight (Y in pounds) and height (X in inches) resulted in the following least squares line: = 140 + 5X. This implies that if the height is increased by 1 inch, the weight is expected to increase on average by 5 pounds.

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line: = 84 +7X. This implies that if advertising is $800, then the predicted amount of sales (in dollars) is $140,000.

A useful graph in almost any regression analysis is a scatterplot of residuals (on the vertical axis) versus fitted values (on the horizontal axis), where a "good" fit not only has small residuals, but it has residuals scattered randomly around zero with no apparent pattern.

In reference to the equation, , the value 0.10 is the expected change in Y per unit change in .

The regression line = 3 + 2X has been fitted to the data points (4, 14), (2, 7), and (1, 4). The sum of the residuals squared will be 8.0.