Question 1

A regression analysis between height y (in cm) and age x (in years) of 2 to 10 years old boys yielded the least squares line $\hat { y }$ = 87 + 6.5x. This implies that by each additional year height is expected to:
A)increase by 93.5cm.
B)increase by 6.5cm.
C)increase by 87cm.
D)decrease by 6.5cm.

Accepted Answer

The coefficient of x (6.5) in the regression equation represents the expected increase in height (y) for each additional year of age (x), hence an increase of 6.5cm per year.

Question 2

In testing the hypotheses: H0: $\rho$s = 0
HA: $\rho$s $\neq$ 0
The Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:
A)2.75.
B)18.178.
C)2.723.
D)17.995.

Accepted Answer

The test statistic for the Spearman rank correlation coefficient, $r_s$, can be calculated using the formula $t = r_s \sqrt{\frac{n-2}{1-r_s^2}}$, where $n$ is the number of observations. Substituting $r_s = 0.389$ and $n = 50$ into the formula gives $t = 0.389 \sqrt{\frac{50-2}{1-0.389^2}} \approx 2.723$.

Question 3

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line $\hat { y }$ = 80 000 + 5x. This implies that an:
A)increase of $1 in advertising is expected to result in an increase of $5 in sales.
B)increase $5 in advertising is expected to result in an increase of $5000 in sales.
C)increase of $1 in advertising is expected to result in an increase of $80 005 in sales.
D)increase of $1 in advertising is expected to result in an increase of $5000 in sales.

Accepted Answer

The answer of A regression analysis between sales (in $1000)...

Question 4

If the standard error of estimate  = 15 and n = 12, then the sum of squares for error, SSE, is:
A)150.
B)180.
C)225.
D)2250.

Accepted Answer

The sum of squares for error (SSE) can be calculated using the formula $SSE = S_{ε}^2 \times (n - 2)$, where $S_{ε}$ is the standard error of estimate and $n$ is the sample size. Plugging in the given values, $SSE = 15^2 \times (12 - 2) = 225 \times 10 = 2250.$

Question 5

In order to estimate with 95% confidence the expected value of y in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?
A)2.228.
B)2.306.
C)1.860.
D)1.812.

Accepted Answer

When the sample size is 10, we need to use the t-distribution with n-2 degrees of freedom. The t-table value for a 95% confidence level and 8 degrees of freedom is 2.306, which is the closest option provided.

Question 6

Given the data points (x,y) = (3,3), (4,4), (5,5), (6,6), (7,7), the least squares estimates of the y-intercept and slope are respectively:
A)0 and 1.
B)-1 and 0.
C)5 and 5.
D)1 and 0.

Accepted Answer

The data points lie perfectly on a line with a slope of 1 and a y-intercept of 0, indicating a direct one-to-one relationship between x and y without any deviation.

Question 7

The standardised residual is defined as:
A)residual divided by the standard error of estimate.
B)residual multiplied by the square root of the standard error of estimate.
C)residual divided by the square of the standard error of estimate.
D)residual multiplied by the standard error of estimate.

Accepted Answer

The standardised residual is the residual divided by the standard error of estimate, which measures the average distance that the observed values fall from the regression line. This value is then standardized to facilitate ease of interpretation and comparison across different models or datasets.

Question 8

The smallest value that the standard error of can assume is:
A)-1.
B)0.
C)1.
D)-2.

Accepted Answer

The standard error of estimate $S _ { ε }$ measures the average distance that the observed values fall from the regression line. It cannot be negative, as it is based on the square root of the squared differences. Therefore, the smallest value it can assume is 0, indicating a perfect fit where all points lie exactly on the regression line.

Question 9

The standard error of estimate, , is given by:
A)SSE/(n - 2).
B) $\sqrt { S S E } / ( n - 2 )$ .
C) $\sqrt { S S E / ( n - 2 ) }$ .
D)SSE/ $\sqrt { n - 2 }$ .

Accepted Answer

The standard error of estimate, $S _ { ε }$, is calculated as the square root of the sum of squared errors (SSE) divided by the degrees of freedom, which is $n - 2$ for simple linear regression. This is represented by option C: $\sqrt { S S E / ( n - 2 ) }$.

Question 10

Which of the following techniques is used to predict the value of one variable on the basis of other variables?
A)Correlation analysis.
B)Coefficient of correlation.
C)Covariance.
D)Regression analysis.

Accepted Answer

Regression analysis is a technique used to predict the value of one variable based on other variables. Correlation analysis, coefficient of correlation, and covariance are all techniques used to measure the relationship between variables, but they do not involve making predictions in the same way that regression analysis does.

Question 11

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?
A)The standard error of estimate.
B)The coefficient of determination.
C)The t-test of the slope.
D)All of these choices are correct.

Accepted Answer

All three options listed can be used to determine whether a linear model should be employed.

The standard error of estimate tells us how much the actual values differ from the predicted values in a linear regression model. A small standard error indicates that the model fits the data well and a linear model may be appropriate.

The coefficient of determination (R-squared) shows the proportion of variation in the dependent variable that is explained by the independent variable(s). A high R-squared value implies that the linear model is a good fit for the data.

The t-test of the slope (also known as the t-test of the coefficient) is used to determine if the slope of the linear regression model is significantly different from zero. If it is significantly different from zero, then the linear model may be appropriate.

Question 12

Which of the following statements is correct when all the actual values of y are on an upward sloping regression line?
A)The coefficient of correlation is equal to one and the sign of the coefficient of correlation will be negative but the sign of the slope with be positive.
B)The coefficient of correlation and the slope must both be equal to 1.
C)The coefficient of correlation is equal to one and the sign of the coefficient of correlation and the sign of the slope will both be positive.
D)The coefficient of correlation and the slope must be equal to - 1.

Accepted Answer

If all the actual values of y are on an upward sloping regression line, it means there is a positive relationship between the independent variable and dependent variable. 
In this case, the coefficient of correlation will be equal to +1 because all the observations are on the regression line, and the sign of the coefficient of correlation and the slope will both be positive because the line is upward sloping. Option A is incorrect because the sign of the coefficient of correlation will also be positive since all the actual values are on the regression line. Option B and D are incorrect because both the coefficient of correlation and slope cannot be equal to 1 or -1 simultaneously.

Question 13

When all the actual values of y and the predicted values of y are equal, the standard error of estimate will be:
A)1.0.
B)-1.0.
C)0.0.
D)2.0.

Accepted Answer

When the actual and predicted values of y are equal, the standard error of estimate is zero because there is no error between the two values.

Question 14

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be:
A)1.0.
B)-1.0.
C)either 1.0 or -1.0.
D)0.0.

Accepted Answer

The coefficient of correlation is 1.0 when all points lie perfectly on a positively sloped line, and -1.0 when all points lie perfectly on a negatively sloped line. Hence, if all points lie on the regression line, the correlation coefficient must be either 1.0 or -1.0, indicating a perfect linear relationship.

Question 15

In regression analysis, if the coefficient of correlation is -1.0, then:
A)the sum of squares for error is -1.0.
B)the sum of squares for regression is 1.0.
C)the sum of squares for error and sum of squares for regression are equal.
D)the sum of squares for regression and total variation in y are equal.

Accepted Answer

A coefficient of correlation of -1.0 indicates a perfect negative linear relationship between the independent and dependent variables. In this case, the sum of squares for regression and total variation in y are equal, as all of the variation in the dependent variable can be explained by the independent variable. The sum of squares for error will be 0, as there is no unexplained variation. Therefore, option D is correct.

Question 16

Which of the following statements best describes why a linear regression is also called a least squares regression model?
A)A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the square of the difference between each actual x data value and the predicted x value.
B)A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the sum of the difference between each actual y data value and the predicted y value.
C)A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the square of each actual y data value and the predicted y value.
D)why a A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the sum of the square of the differences between each actual y data value and the predicted y value.

Accepted Answer

In the least squares regression model, the regression line is calculated by minimizing the sum of the square of the differences between each actual y data value and the predicted y value. Therefore, statement D is the correct answer. A, B, and C are not correct because they do not accurately describe the method used to calculate the regression line in a least squares regression model.

Question 17

If the coefficient of correlation between x and y is close to −1.0, which of the following statements is correct?
A)There is a strong, positive linear relationship between x and y, where there may or may not be any causal relationship between x and y.
B)There is a strong, negative linear relationship between x and y, where there must be a causal relationship between x and y.
C)There is a weak, negative linear relationship between x and y, where there may or may not be any causal relationship between x and y.
D)There is a strong, negative linear relationship between x and y, where there may or may not be any causal relationship between x and y.

Accepted Answer

A coefficient of correlation of −1.0 indicates a perfect negative linear relationship between x and y, meaning that as x increases, y decreases at a constant rate. This suggests a strong negative linear relationship between x and y. However, correlation does not imply causation, so it is possible that there may or may not be a causal relationship between x and y. Therefore, the correct statement is that there is a strong, negative linear relationship between x and y, where there may or may not be any causal relationship between x and y.

Question 18

If the coefficient of determination is 81%, and the linear regression model has a negative slope, what is the value of the coefficient of correlation?
A)− 0.81
B)0.90
C)0.81
D)−0.90

Accepted Answer

The coefficient of correlation, denoted by "r," is the square root of the coefficient of determination (r = sqrt(R2)). Since the coefficient of determination is 81%, then the coefficient of correlation is sqrt(0.81) = 0.9. Additionally, the fact that the linear regression model has a negative slope indicates that there is a negative correlation between the variables, which is reflected in the negative sign of the slope. Therefore, the best choice is D, −0.90.

Question 19

Of the values of the coefficient of determination listed below, which one implies the greatest value of the sum of squares for regression, given that the total variation in y is 1800?
A)0.69.
B)0.96.
C)96.
D)-100.

Accepted Answer

The coefficient of determination (R-squared) is the proportion of the variance in the dependent variable (y) that is predictable from the independent variable (x). The sum of squares for regression measures the amount of variance in y that can be explained by the independent variable. It is calculated as the difference between the total sum of squares (SS Total) and the sum of squares error (SS Error). Thus, the greater the R-squared value, the greater the sum of squares for regression. Of the given choices, only B has a value greater than 0.69, implying the highest sum of squares for regression. Therefore, B is the best choice.

Question 20

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line $\hat { y }$ = 75 +6x. This implies that if $800 is spent on advertising, then the predicted amount of sales (in dollars) is:
A)$4875.
B)$123 000.
C)$487 500.
D)$12 300.

Accepted Answer

The equation $\hat{y} = 75 + 6x$ is given in terms of sales in $1000 and advertising in $100. For $800 spent on advertising, x = 800/100 = 8. Plug x = 8 into the equation: $\hat{y} = 75 + 6(8) = 75 + 48 = 123$. Since $\hat{y}$ is in $1000, the predicted sales is $123,000.

Quiz 15: Simple Linear Regression and Correlation

A regression analysis between height y (in cm) and age x (in years) of 2 to 10 years old boys yielded the least squares line $\hat { y }$ = 87 + 6.5x. This implies that by each additional year height is expected to:

In testing the hypotheses: H0: $\rho$ s = 0 HA: $\rho$ s $\neq$ 0 The Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line $\hat { y }$ = 80 000 + 5x. This implies that an:

If the standard error of estimate = 15 and n = 12, then the sum of squares for error, SSE, is:

In order to estimate with 95% confidence the expected value of y in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

Given the data points (x,y) = (3,3) , (4,4) , (5,5) , (6,6) , (7,7) , the least squares estimates of the y-intercept and slope are respectively:

The standardised residual is defined as:

The smallest value that the standard error of estimatecan assume is:

The standard error of estimate, $The standard error of estimate, , is given by: A) SSE/(n - 2) . B) \sqrt { S S E } / ( n - 2 ) . C) \sqrt { S S E / ( n - 2 ) } . D) SSE/ \sqrt { n - 2 } .$ , is given by:

Which of the following techniques is used to predict the value of one variable on the basis of other variables?

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?

Which of the following statements is correct when all the actual values of y are on an upward sloping regression line?

When all the actual values of y and the predicted values of y are equal, the standard error of estimate will be:

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

In regression analysis, if the coefficient of correlation is -1.0, then:

Which of the following statements best describes why a linear regression is also called a least squares regression model?

If the coefficient of correlation between x and y is close to −1.0, which of the following statements is correct?

If the coefficient of determination is 81%, and the linear regression model has a negative slope, what is the value of the coefficient of correlation?

Of the values of the coefficient of determination listed below, which one implies the greatest value of the sum of squares for regression, given that the total variation in y is 1800?

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line $\hat { y }$ = 75 +6x. This implies that if $800 is spent on advertising, then the predicted amount of sales (in dollars) is: