Question 1

In a regression analysis, if r² = 1, then _____. A) SSE = SST B) SSE = 1 C) SSR = SSE D) SSR = SST

Accepted Answer

When $r^2 = 1$, it indicates a perfect linear relationship between the independent and dependent variables. This means all the variation in the dependent variable is explained by the independent variable, hence the sum of squares due to regression (SSR) equals the total sum of squares (SST), indicating no variation is left unexplained (SSE = 0).

Question 2

If there is a very strong correlation between two variables, then the coefficient of correlation must be _____.
A) much larger than 1, if the correlation is positive
B) much smaller than 1, if the correlation is negative
C) either much larger than 1 or much smaller than 1
D) None of the answers is correct.

Accepted Answer

The coefficient of correlation ranges from -1 to 1, where 1 indicates a perfect positive correlation and -1 indicates a perfect negative correlation. It cannot be larger than 1 or smaller than -1.

Question 3

A regression analysis between sales (in $1000s) and price (in dollars) resulted in the following equation: ? $\hat { y}$ = 50,000 ? 8x
?
The above equation implies that an increase of _____.
A) $1 in price is associated with a decrease of $8 in sales
B) $8 in price is associated with an increase of $8,000 in sales
C) $1 in price is associated with a decrease of $42,000 in sales
D) $1 in price is associated with a decrease of $8,000 in sales

Accepted Answer

The coefficient of x (-8) indicates the change in the dependent variable (sales, in $1000s) for a one-unit change in the independent variable (price, in dollars). Therefore, a $1 increase in price is associated with a decrease of $8,000 in sales (since the unit of sales is in $1000s, -8 translates to -$8,000).

Question 4

If the coefficient of determination is a positive value, then the regression equation _____.
A) must have a positive slope
B) must have a negative slope
C) could have either a positive or a negative slope
D) must have a positive y-intercept

Accepted Answer

The coefficient of determination, denoted as R^2, measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It does not indicate the direction of the relationship (positive or negative slope) between the variables, only the strength of the association. Therefore, the regression equation could have either a positive or a negative slope.

Question 5

If a data set has SST = 2,000 and SSE = 800, then the coefficient of determination is _____.
A) .4
B) .6
C) .5
D) .8

Accepted Answer

The coefficient of determination, denoted as $R^2$, is calculated as $1 - \frac{SSE}{SST}$, where $SSE$ is the sum of squares due to error and $SST$ is the total sum of squares. Plugging in the given values, $R^2 = 1 - \frac{800}{2000} = 1 - 0.4 = 0.6$.

Question 6

In simple linear regression, r² is the _____. A) estimated regression equation B) coefficient of correlation C) sum of the squared residuals D) coefficient of determination

Accepted Answer

r², or the coefficient of determination, measures the proportion of the variance in the dependent variable that is predictable from the independent variable in a regression model.

Question 7

If the coefficient of correlation is .4, the percentage of variation in the dependent variable explained by the estimated regression equation _____.
A) is 40%
B) is 16%
C) is 4%
D) can be any positive value

Accepted Answer

The percentage of variation in the dependent variable explained by the estimated regression equation is found by squaring the coefficient of correlation (r^2). In this case, 0.4^2 = 0.16, which means 16%.

Question 8

Which of the following is correct? A) SSE = SSR + SST B) SSR = SSE + SST C) SST = SSR + SSE D) SST = (SSR)²

Accepted Answer

SST (Total Sum of Squares) equals the sum of the squares of the deviations of all the data points from their mean. This can be further broken down into two components: SSR (Regression Sum of Squares) and SSE (Error Sum of Squares). SSR measures how much of the variance in the data is explained by the regression model while SSE measures the unexplained variance or error. Therefore, the relationship between these three values can be expressed as SST = SSR + SSE. Option A and B are incorrect as they reverse the correct relationship between SSE and SSR. Option D is unrelated to the concept of sum of squares in regression analysis.

Question 9

A regression analysis between sales (y in $1000) and advertising (x in dollars) resulted in the following equation: $\hat { y }$ = 50,000 + 6x ?
The above equation implies that an increase of _____.
A) $6 in advertising is associated with an increase of $6,000 in sales
B) $1 in advertising is associated with an increase of $6 in sales
C) $1 in advertising is associated with an increase of $56,000 in sales
D) $1 in advertising is associated with an increase of $6,000 in sales

Accepted Answer

The slope of the regression line is 6, which means that an increase of $1 in advertising is associated with an increase of $6 in sales.

Question 10

It is possible for the coefficient of determination to be _____.
A) larger than 1
B) less than 1
C) less than 0
D) All of the answers are correct, depending on the situation under consideration.

Accepted Answer

The coefficient of determination, also known as R-squared, measures the proportion of variation in the dependent variable that is explained by the independent variable(s). It is always between 0 and 1, inclusive, with a value of 1 indicating a perfect fit of the data to the model. Therefore, it is not possible for the coefficient of determination to be larger than 1 or less than 0. However, it is possible for the coefficient of determination to be less than 1 but greater than 0, indicating that the model explains only a portion of the variation in the data.

Question 11

If the coefficient of determination is equal to 1, then the coefficient of correlation _____.
A) must also be equal to 1
B) can be either -1 or 1
C) can be any value between -1 and 1
D) must be -1

Accepted Answer

The coefficient of determination (R^2) being equal to 1 indicates a perfect linear relationship between the variables. The coefficient of correlation (r) describes the direction of this relationship, which can be perfectly positive (r = 1) or perfectly negative (r = -1) when R^2 is 1.

Question 12

If all the points of a scatter diagram lie on the least squares regression line, then the coefficient of determination for these variables based on these data _____.
A) is 0
B) is 1
C) is either 1 or -1, depending upon whether the relationship is positive or negative
D) could be any value between -1 and 1

Accepted Answer

If all the points lie on the least squares regression line, it means that the line explains all the variation in the data. Thus, the coefficient of determination (R-squared) would be 1, indicating that 100% of the variation in the dependent variable can be explained by the independent variable(s).

Question 13

In regression and correlation analysis, if SSE and SST are known, then with this information the _____.
A) coefficient of determination can be computed
B) slope of the line can be computed
C) y-intercept can be computed
D) All of the answers are correct.

Accepted Answer

The coefficient of determination (R-squared) can be calculated using the formula R-squared = 1 - (SSE/SST), where SSE is the sum of squared errors and SST is the total sum of squares. The other options (B and C) require the knowledge of additional information such as the slope and y-intercept of the regression line, respectively.

Question 14

If the coefficient of correlation is .8, then the percentage of variation in the dependent variable explained by the estimated regression equation is _____.
A) 0.80%
B) 80%
C) 0.64%
D) 64%

Accepted Answer

The percentage of variation in the dependent variable explained by the estimated regression equation is found by squaring the coefficient of correlation (r^2). Thus, 0.8^2 = 0.64 or 64%.

Question 15

In a regression analysis, if r² = 1, then _____. A) SSE must also be equal to 1 B) SSE must be equal to 0 C) SSE can be any positive value D) SSE must be negative

Accepted Answer

In a regression analysis, r² represents the correlation coefficient between the predicted values and the actual values. If it is equal to 1, it means that the predicted values perfectly fit the actual values, resulting in an SSE (sum of squared errors) of 0. Therefore, the answer is B, SSE must be equal to 0.

Question 16

In a regression analysis, if SSE = 200 and SSR = 300, then the coefficient of determination is _____.
A) .667
B) .600
C) .400
D) 1.500

Accepted Answer

The answer of In a regression analysis, if SSE =...

Question 17

In a regression analysis, if SST = 4500 and SSE = 1575, then the coefficient of determination is _____.
A) .35
B) .65
C) 2.85
D) .45

Accepted Answer

The coefficient of determination (R-squared) is calculated as R-squared = (SST - SSE) / SST. Plugging in the given values, we get R-squared = (4500 - 1575) / 4500 = 0.65. Therefore, the answer is Choice B.

Question 18

In a regression analysis, if SSE = 500 and SSR = 300, then the coefficient of determination is _____.
A) .600
B) .166
C) 1.666
D) .375

Accepted Answer

The coefficient of determination (R-squared) is defined as the proportion of the total variation in the dependent variable (SST) that is explained by the regression model (SSR). 
R-squared = SSR / SST
SSE + SSR = SST
Hence, SST =500+300 = 800
R-squared = 300/800 = 0.375
Therefore, the coefficient of determination is 0.375 or 37.5%. The answer is D.

Question 19

SSE can never be _____.
A) larger than SST
B) smaller than SST
C) equal to 1
D) equal to 0

Accepted Answer

SSE (Sum of Squares due to Error) can never be larger than SST (Total Sum of Squares) because SST is the total variability in the data, and SSE is a portion of that variability not explained by the model.

Question 20

Larger values of r² imply that the observations are more closely grouped about the _____. A) average value of the independent variables B) average value of the dependent variable C) least squares line D) origin

Accepted Answer

The correlation coefficient r² measures the strength and direction of the linear relationship between the independent variable (S1) and the dependent variable (U1). Larger values of r² indicate a stronger linear relationship, which means the observations are more closely grouped about the least squares line. Therefore, the best choice is C.

Quiz 14: Simple Linear Regression

In a regression analysis, if r2 = 1, then _____.

If there is a very strong correlation between two variables, then the coefficient of correlation must be _____.

A regression analysis between sales (in $1000s) and price (in dollars) resulted in the following equation: ? y^\hat { y}y^​ = 50,000 ? 8x ? The above equation implies that an increase of _____.

If the coefficient of determination is a positive value, then the regression equation _____.

If a data set has SST = 2,000 and SSE = 800, then the coefficient of determination is _____.

In simple linear regression, r2 is the _____.

If the coefficient of correlation is .4, the percentage of variation in the dependent variable explained by the estimated regression equation _____.

Which of the following is correct?

A regression analysis between sales (y in $1000) and advertising (x in dollars) resulted in the following equation: y^\hat { y }y^​ = 50,000 + 6x ? The above equation implies that an increase of _____.

It is possible for the coefficient of determination to be _____.

If the coefficient of determination is equal to 1, then the coefficient of correlation _____.

If all the points of a scatter diagram lie on the least squares regression line, then the coefficient of determination for these variables based on these data _____.

In regression and correlation analysis, if SSE and SST are known, then with this information the _____.

If the coefficient of correlation is .8, then the percentage of variation in the dependent variable explained by the estimated regression equation is _____.

In a regression analysis, if r2 = 1, then _____.

In a regression analysis, if SSE = 200 and SSR = 300, then the coefficient of determination is _____.

In a regression analysis, if SST = 4500 and SSE = 1575, then the coefficient of determination is _____.

In a regression analysis, if SSE = 500 and SSR = 300, then the coefficient of determination is _____.

SSE can never be _____.

Larger values of r2 imply that the observations are more closely grouped about the _____.

In a regression analysis, if r² = 1, then _____.

A regression analysis between sales (in $1000s) and price (in dollars) resulted in the following equation: ? $\hat { y}$ = 50,000 ? 8x ? The above equation implies that an increase of _____.

In simple linear regression, r² is the _____.

A regression analysis between sales (y in $1000) and advertising (x in dollars) resulted in the following equation: $\hat { y }$ = 50,000 + 6x ? The above equation implies that an increase of _____.

In a regression analysis, if r² = 1, then _____.

Larger values of r² imply that the observations are more closely grouped about the _____.