Question 1

The regression line 
 = 2 + 3x has been fitted to the data points (4,11), (2,7), and (1,5). The residual sum of squares will be 10.0.

Accepted Answer

The answer of The regression line 
 = 2 +...

Question 2

In a simple linear regression model, if r² is 0.75, then 75% of the variation in the dependent variable y can be explained by the regression line, on the independent variable x.

Accepted Answer

This is correct as the correlation coefficient (r) in a simple linear regression model represents the strength and direction of the linear relationship between the independent variable x and dependent variable y. Square of the correlation coefficient (r²) represents the proportion of variation in y that can be explained by the regression line on x. Therefore, 0.75 (r²)² = 0.5625, which means that 56.25% of the variation in y can be explained by the regression line on x.

Question 3

In a regression problem the following pairs (x, y) are given: (3,-2), (3,-1), (3,0), (3,1) and (3,2). This indicates that the coefficient of correlation is -1.

Accepted Answer

The given pairs (x, y) have constant x value, which indicates that there is no variation in x and thus no correlation between x and y. The correlation cannot be -1 in this case.

Question 4

If the coefficient of correlation is -0.7, then the percentage of the variation in y that is explained by the regression line is 70%.

Accepted Answer

The coefficient of correlation measures the strength and direction of the linear relationship between two variables, but it does not tell us the percentage of variation in y that is explained by the regression line. The percentage of variation in the dependent variable y that is explained by the regression line is given by the coefficient of determination (r^2), which is the square of the coefficient of correlation. Therefore, if the coefficient of correlation is -0.7, the percentage of variation in y explained by the regression line is r^2 = (-0.7)^2 = 0.49 or 49%.

Question 5

We standardise residuals in the same way that we standardise all variables, by subtracting the mean and dividing by the variance.

Accepted Answer

We standardise residuals by dividing them by their estimated standard deviation, not by the variance.

Question 6

The value of the sum of squares for regression, SSR, can never be larger than the value of sum of squares for error, SSE.

Accepted Answer

The value of the sum of squares for regression (SSR) can indeed be larger than the sum of squares for error (SSE) because SSR reflects the variation explained by the regression model, while SSE represents the unexplained variation. The total variation in the data (SST) is the sum of SSR and SSE, and there's no inherent constraint that SSR must be smaller than SSE.

Question 7

A regression analysis between weight $$y$$ (in kilograms) and height $$x$$ (in centimetres) yielded the least squares line 
 = 135 + 6 $$x$$ . We estimate that for each 1 cm in height, that weight will cedrease by 6 kilograms, on average.

Accepted Answer

The statement in the question should be reversed. The least squares line indicates that weight will increase by 6 kilograms for each 1 cm increase in height.

Question 8

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be +1.0.

Accepted Answer

The answer of If all the points in a scatter...

Question 9

When the actual values y of a dependent variable and the corresponding predicted values 
 are the same, the standard error of estimate, $$S _ { \varepsilon }$$ , will be -1.0.

Accepted Answer

The standard error of estimate can never be negative. It is always a positive value that represents the average amount by which the predicted values differ from the actual values.

Question 10

Statisticians have shown that the sample y-intercept $$b _ { 0 }$$ and sample slope coefficient $$b _ { 1 }$$ are unbiased estimators of the population regression parameters $$\beta _ { 0 }$$ and $$\beta _ { 1 }$$ .

Accepted Answer

This statement is true. The sample y-intercept and sample slope coefficient are calculated using formulas that take into account the entire population, thereby ensuring that they are unbiased estimators of the population regression parameters.

Question 11

In a simple linear regression problem, the least squares line is 
 = -3.75 + 1.25 $$x$$ , and the coefficient of determination is 0.81. The coefficient of correlation must be 0.90.

Accepted Answer

The answer of In a simple linear regression problem, the...

Question 12

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line y-hat = 77 +8x. This implies that if advertising is $600, then the predicted amount of sales (in dollars) is $4877.

Accepted Answer

The predicted sales should be calculated as y-hat = 77 + 8(600/100) = 77 + 48 = 125 (in $1000), which is $125,000, not $4877.

Question 13

The standard error of the estimate is the standard deviation of the error variable.

Accepted Answer

This is correct. The standard error of the estimate is also known as the standard deviation of the residuals, which are the differences between the predicted values from a regression model and the actual values. It measures the spread of the errors around the regression line.

Question 14

The value of the sum of squares for regression, SSR, can never be equal to the value of total sum of squares, SST.

Accepted Answer

The value of SSR can be equal to SST if the regression model fits the data perfectly, in which case R-squared would be equal to 1.

Question 15

If the standard error of estimate $$S _ { \varepsilon }$$ = 20 and n = 8, then the sum of squares for error, SSE, is 2400.

Accepted Answer

We know that the formula for calculating SSE is:
SSE = (n-2) * S^2e
where S^2e is the variance of the errors.

Given:
S^2e = (S_e)^2 = (20)^2 = 400
n = 8

Substituting the values in the SSE formula, we get:
SSE = (8-2) * 400 = 2400

Therefore, the statement is true.

Question 16

In order to estimate with 95% confidence a particular value of $$y$$ for a given value of $$x$$ in a simple linear regression problem, a random sample of 20 observations is taken. The appropriate table value that would be used is 2.101.

Accepted Answer

True. The appropriate table value for a 95% confidence level with 18 degrees of freedom is 2.101, which can be found in a t-distribution table or calculated using statistical software.

Question 17

The residuals are observations of the error variable $$\varepsilon$$ . Consequently, the minimised square of deviations is called the sum of squares for error, denoted SSE.

Accepted Answer

The answer of The residuals are observations of the error...

Question 18

The residual $$r _ { i }$$ is defined as the difference between the actual value $$y _ { i }$$ and the estimated value 
 .

Accepted Answer

This is the correct definition of the residual.

Question 19

A regression analysis between y, sales (in $1000) and x, advertising (in $) yielded the least squares line 
 = 60 + 5x. We can interpret the slope by saying that we estimate for each extra $1 spent on advertising that sales will increase by $5 000, on average.

Accepted Answer

The answer of A regression analysis between y, sales (in...

Question 20

Correlation analysis is used to determine the strength of a non-linear relationship between an independent variable x and a dependent variable y.

Accepted Answer

Correlation analysis is used to determine the strength and direction of a linear relationship between an independent variable x and a dependent variable y. Non-linear relationships are analyzed using other statistical methods such as regression analysis.

Quiz 18: Simple Linear Regression and Correlation

The regression line = 2 + 3x has been fitted to the data points (4,11), (2,7), and (1,5). The residual sum of squares will be 10.0.

In a simple linear regression model, if r² is 0.75, then 75% of the variation in the dependent variable y can be explained by the regression line, on the independent variable x.

In a regression problem the following pairs (x, y) are given: (3,-2), (3,-1), (3,0), (3,1) and (3,2). This indicates that the coefficient of correlation is -1.

If the coefficient of correlation is -0.7, then the percentage of the variation in y that is explained by the regression line is 70%.

We standardise residuals in the same way that we standardise all variables, by subtracting the mean and dividing by the variance.

The value of the sum of squares for regression, SSR, can never be larger than the value of sum of squares for error, SSE.

A regression analysis between weight $y$ (in kilograms) and height $x$ (in centimetres) yielded the least squares line = 135 + 6 $x$ . We estimate that for each 1 cm in height, that weight will cedrease by 6 kilograms, on average.

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be +1.0.

Statisticians have shown that the sample y-intercept $b _ { 0 }$ and sample slope coefficient $b _ { 1 }$ are unbiased estimators of the population regression parameters $\beta _ { 0 }$ and $\beta _ { 1 }$ .

In a simple linear regression problem, the least squares line is = -3.75 + 1.25 $x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be 0.90.

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line y-hat = 77 +8x. This implies that if advertising is $600, then the predicted amount of sales (in dollars) is $4877.

The standard error of the estimate is the standard deviation of the error variable.

The value of the sum of squares for regression, SSR, can never be equal to the value of total sum of squares, SST.

If the standard error of estimate $S _ { \varepsilon }$ = 20 and n = 8, then the sum of squares for error, SSE, is 2400.

In order to estimate with 95% confidence a particular value of $y$ for a given value of $x$ in a simple linear regression problem, a random sample of 20 observations is taken. The appropriate table value that would be used is 2.101.

The residuals are observations of the error variable $\varepsilon$ . Consequently, the minimised square of deviations is called the sum of squares for error, denoted SSE.

The residual $r _ { i }$ is defined as the difference between the actual value $y _ { i }$ and the estimated value $The residual r _ { i } is defined as the difference between the actual value y _ { i } and the estimated value .$ .

A regression analysis between y, sales (in $1000) and x, advertising (in $) yielded the least squares line = 60 + 5x. We can interpret the slope by saying that we estimate for each extra $1 spent on advertising that sales will increase by $5 000, on average.

Correlation analysis is used to determine the strength of a non-linear relationship between an independent variable x and a dependent variable y.

Quiz 18: Simple Linear Regression and Correlation

The regression line = 2 + 3x has been fitted to the data points (4,11), (2,7), and (1,5). The residual sum of squares will be 10.0.

In a simple linear regression model, if r2 is 0.75, then 75% of the variation in the dependent variable y can be explained by the regression line, on the independent variable x.

In a regression problem the following pairs (x, y) are given: (3,-2), (3,-1), (3,0), (3,1) and (3,2). This indicates that the coefficient of correlation is -1.

If the coefficient of correlation is -0.7, then the percentage of the variation in y that is explained by the regression line is 70%.

We standardise residuals in the same way that we standardise all variables, by subtracting the mean and dividing by the variance.

The value of the sum of squares for regression, SSR, can never be larger than the value of sum of squares for error, SSE.

A regression analysis between weight yyy (in kilograms) and height xxx (in centimetres) yielded the least squares line = 135 + 6 xxx . We estimate that for each 1 cm in height, that weight will cedrease by 6 kilograms, on average.

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be +1.0.

When the actual values y of a dependent variable and the corresponding predicted values are the same, the standard error of estimate, SεS _ { \varepsilon }Sε​ , will be -1.0.

Statisticians have shown that the sample y-intercept b0b _ { 0 }b0​ and sample slope coefficient b1b _ { 1 }b1​ are unbiased estimators of the population regression parameters β0\beta _ { 0 }β0​ and β1\beta _ { 1 }β1​ .

In a simple linear regression problem, the least squares line is = -3.75 + 1.25 xxx , and the coefficient of determination is 0.81. The coefficient of correlation must be 0.90.

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line y-hat = 77 +8x. This implies that if advertising is $600, then the predicted amount of sales (in dollars) is $4877.

The standard error of the estimate is the standard deviation of the error variable.

The value of the sum of squares for regression, SSR, can never be equal to the value of total sum of squares, SST.

If the standard error of estimate SεS _ { \varepsilon }Sε​ = 20 and n = 8, then the sum of squares for error, SSE, is 2400.

In order to estimate with 95% confidence a particular value of yyy for a given value of xxx in a simple linear regression problem, a random sample of 20 observations is taken. The appropriate table value that would be used is 2.101.

The residuals are observations of the error variable ε\varepsilonε . Consequently, the minimised square of deviations is called the sum of squares for error, denoted SSE.

The residual rir _ { i }ri​ is defined as the difference between the actual value yiy _ { i }yi​ and the estimated value .

A regression analysis between y, sales (in $1000) and x, advertising (in $) yielded the least squares line = 60 + 5x. We can interpret the slope by saying that we estimate for each extra $1 spent on advertising that sales will increase by $5 000, on average.

Correlation analysis is used to determine the strength of a non-linear relationship between an independent variable x and a dependent variable y.

In a simple linear regression model, if r² is 0.75, then 75% of the variation in the dependent variable y can be explained by the regression line, on the independent variable x.

A regression analysis between weight $y$ (in kilograms) and height $x$ (in centimetres) yielded the least squares line = 135 + 6 $x$ . We estimate that for each 1 cm in height, that weight will cedrease by 6 kilograms, on average.

Statisticians have shown that the sample y-intercept $b _ { 0 }$ and sample slope coefficient $b _ { 1 }$ are unbiased estimators of the population regression parameters $\beta _ { 0 }$ and $\beta _ { 1 }$ .

In a simple linear regression problem, the least squares line is = -3.75 + 1.25 $x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be 0.90.

If the standard error of estimate $S _ { \varepsilon }$ = 20 and n = 8, then the sum of squares for error, SSE, is 2400.

In order to estimate with 95% confidence a particular value of $y$ for a given value of $x$ in a simple linear regression problem, a random sample of 20 observations is taken. The appropriate table value that would be used is 2.101.

The residuals are observations of the error variable $\varepsilon$ . Consequently, the minimised square of deviations is called the sum of squares for error, denoted SSE.

The residual $r _ { i }$ is defined as the difference between the actual value $y _ { i }$ and the estimated value $The residual r _ { i } is defined as the difference between the actual value y _ { i } and the estimated value .$ .