Question 1

If the standard error of estimate 5_yx = 18 and n = 10, then the error sum of squares, SSE, is: A)2916. B)2592. C)1800. D)3240.

Accepted Answer

The formula for the error sum of squares (SSE) is: 
SSE = (n-2) * MSE 
where MSE is the mean square error, which is equal to the variance of the errors divided by the sample size: 
MSE = s^2 = (1/n) * Σ(ei - ȳ)^2

We are given the standard error of estimate (s), which is equal to the square root of MSE:
s = √(MSE)

Using the given values, s = 18, n = 10, we can solve for MSE: 
s^2 = MSE/n 
18^2 = MSE/10 
MSE = 3240/10 
MSE = 324

Finally, we can calculate SSE: 
SSE = (n-2) * MSE 
SSE = (10-2) * 324 
SSE = 8 * 324 
SSE = 2592

Therefore, the answer is B: 2592.

Question 2

The following values are listed as coefficients of correlation (r).The one that indicates an inverse relationship between the two variables x and y is:
A) 0.0.
B) -0.8.
C) 0.9.
D) 1.3.
E) -1.4.

Accepted Answer

A negative correlation coefficient (r) indicates an inverse relationship between two variables, and -0.8 is a strong negative correlation. The other options are positive correlation coefficients, indicating a direct relationship between the variables, or invalid coefficients outside of the -1 to 1 range.

Question 3

For a given value of x,the estimation interval for an individual y observation is called the:
A) confidence interval.
B) residual.
C) prediction interval.
D) least-squares interval.
E) standard error of estimate.

Accepted Answer

The prediction interval is used to estimate the range of values within which a future observation is expected to fall with a certain level of confidence. This interval takes into account the variation in both the explanatory variable (x) and the response variable (y). The confidence interval (A) is used to estimate the range of values within which the mean of the y variable is expected to fall, while the residual (B) measures the difference between the observed and predicted values of y. The least-squares interval (D) and standard error of estimate (E) are not commonly used terms in inferential statistics. Therefore, the correct answer is C) prediction interval.

Question 4

Correlation analysis requires that the scales of measurement be expressed in either:
A) nominal or ordinal.
B) ordinal or ratio.
C) interval or ratio.
D) nominal or ratio.
E) nominal or interval.

Accepted Answer

Correlation analysis requires data to be measured on an interval or ratio scale. Nominal and ordinal scales only provide information on categories or rankings and do not have a meaningful numerical distance between values. Interval scales have a meaningful distance between values but do not have a true zero point, while ratio scales have a true zero point and a meaningful distance between values. Therefore, correlation analysis can only be performed on variables measured on interval or ratio scales.

Question 5

In simple linear regression,the coefficient of correlation r and the least squares estimate $b _ { 1 }$ of the population slope $\beta _ { 1 }$ :
A) must have the same numerical value.
B) must have opposite signs.
C) must have the same sign.
D) may have opposite signs or the same sign.

Accepted Answer

The coefficient of correlation $r$ and the least squares estimate $b_1$ of the population slope $\beta_1$ must have the same sign. This is because both $r$ and $b_1$ reflect the direction of the linear relationship between the two variables. If the relationship is positive, both will be positive; if the relationship is negative, both will be negative.

Question 6

Which of the following table values would be appropriate for a 95% confidence interval for the mean of y from a simple linear regression problem if the sample size is 7?
A) 1.895
B) 2.015
C) 2.365
D) 2.571
E) 1.960

Accepted Answer

For a 95% confidence interval in a simple linear regression with a sample size of 7, you use the t-distribution with n-2 degrees of freedom (5 in this case). The t-value for 5 degrees of freedom at 95% confidence is approximately 2.571.

Question 7

Given the least squares regression line $\hat{y}$ = -2.88 + 1.77x,and a coefficient of determination of 0.81,the coefficient of correlation is:
A) -0.88.
B) +0.88.
C) +0.90.
D) -0.90.

Accepted Answer

The coefficient of correlation is the square root of the coefficient of determination, so we have $\sqrt{0.81} = 0.9$. The positive value indicates a positive linear relationship between $x$ and $y$, and the magnitude indicates a strong relationship. Therefore, the answer is (C).

Question 8

The regression line   $\hat{y}$= 3 + 2x has been fitted to the data points (4,8), (2,5), and (1,2). The residual sum of squares will be:
A)	10.
B)	15.
C)	13.
D)	22.

Accepted Answer

The answer of The regression line   $\hat{y}$= 3...

Question 9

The number of degrees of freedom associated with the standard error of estimate is
A) n-1 since only the slope is estimated from sample data.
B) n-1 since only the intercept is estimated from sample data.
C) n-1 since only the predicted value of y is estimated from sample data.
D) n-2 since the slope and the intercept are estimated from sample data.

Accepted Answer

The standard error of estimate is the standard deviation of the residuals, which are the differences between the observed values of y and the predicted values of y. To estimate the slope and intercept, we use the sample data, which reduces the degrees of freedom by two. Therefore, the number of degrees of freedom associated with the standard error of estimate is n-2.

Question 10

For the values of the coefficient of determination listed below,which one implies the greatest value of SSR (regression sum of squares)given that SST = 500?
A) 0.95
B) -1.00
C) 0.28
D) 0.00
E) -0.88

Accepted Answer

The coefficient of determination, R-squared, is defined as the ratio of regression sum of squares (SSR) to the total sum of squares (SST). Thus, if SST = 500, the SSR will be highest when R-squared is highest. 
Out of the given options, only A (0.95) represents the highest possible R-squared value (since R-squared is always between 0 and 1). Therefore, A implies the greatest value of SSR.

Question 11

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

Since we want to estimate the expected value of y with 95% confidence, this is a two-tailed test with alpha = 0.05/2 = 0.025. With a sample size of 10, we have 10-2=8 degrees of freedom. Looking at a t-table, the value closest to the one-tailed 0.025 critical value for 8 degrees of freedom is 2.306, which is provided as choice B. Therefore, we would use this value to estimate the expected value of y with 95% confidence.

Question 12

In a regression problem the following pairs of (x,y)are given: (2,1),(2,-1),(2,0),(2,-2)and (2,2).That indicates that the:
A) coefficient of correlation is -1.
B) coefficient of correlation is 0.
C) coefficient of correlation is 1.
D) coefficient of determination is between -1 and 1.

Accepted Answer

Since all x values are the same and y values have no linear relationship with x, the coefficient of correlation is 0.

Question 13

In publishing the results of some research work,the following values of the coefficient of determination were listed.Which one would appear to be incorrect?
A) 0.91
B) 0.06
C) 0.47
D) -0.64
E) 0.00

Accepted Answer

The coefficient of determination, also known as R-squared, is a value between 0 and 1 that represents the proportion of the variance in the dependent variable that is explained by the independent variable(s). A negative coefficient of determination is not possible, so choice D (-0.64) appears to be incorrect.

Question 14

All of the values of an independent variable equal the same number.Regressing a dependent variable on this independent variable will result in a coefficient of determination (r² )of: A) 0.0. B) -1.0. C) 2.3. D) -2.3. E) 1.0.

Accepted Answer

When all the values of an independent variable equal the same number, this means there is no variation in the independent variable. Therefore, the dependent variable cannot be predicted by the independent variable, resulting in an r-squared value of 0.0.

Question 15

The vertical spread of the data points about the regression line is measured by the:
A) regression coefficient.
B) standard error of estimate.
C) y-intercept.
D) homoscedasticity coefficient.
E) t-ratio.

Accepted Answer

The standard error of estimate (also known as the standard deviation of the residuals) measures the spread of the data points around the regression line. It is the most common measure of the vertical spread of the data points in regression analysis. The other options mentioned do not measure the spread of data points around the regression line.

Question 16

An indication of no linear relationship between two variables would be:
A) a coefficient of determination equal to 1.
B) a coefficient of determination equal to -1.
C) a coefficient of correlation of 0.
D) a coefficient of correlation equal to -1.
E) Both "A" and "B" are correct.

Accepted Answer

A coefficient of correlation of 0 indicates no linear relationship between the two variables. A coefficient of determination equal to 1 or -1 indicates a perfect positive or negative linear relationship, respectively, while a coefficient of determination between 0 and 1 indicates the proportion of the variation in one variable that can be explained by the other variable. Therefore, choices A, B, and D are incorrect in this scenario.

Question 17

The value for SSE equals zero.This means that the coefficient of determination (r² )must equal: A) 0.0. B) -1.0. C) 2.3. D) -2.3. E) 1.0.

Accepted Answer

The coefficient of determination (r²) represents the proportion of variation in the dependent variable (Y) that can be explained by the independent variable (X). When SSE equals zero, it means that there is no variation in the dependent variable that is unaccounted for by the independent variable. Therefore, all of the variation in Y can be explained by X, and the coefficient of determination equals 1.0 (or 100%).

Question 18

If all the points in a scatter diagram lie on the least squares regression line,then the coefficient of correlation:
A) must be 1.0.
B) must be -1.0.
C) must be either 1.0 or -1.0.
D) must be 0.

Accepted Answer

If all the points in a scatter diagram lie on the least squares regression line, then the relationship between the variables is perfectly linear, and the coefficient of correlation must be either 1.0 or -1.0. A correlation coefficient of 1.0 indicates a perfect positive correlation, while a correlation coefficient of -1.0 indicates a perfect negative correlation.

Question 19

Correlation analysis is used to determine the:
A) strength of the relationship between x and y.
B) least squares estimates of the regression parameters.
C) predicted value of y for a given value of x.
D) coefficient of determination.

Accepted Answer

Correlation analysis is used to determine the strength of the relationship between two variables, typically denoted as x and y. It does not provide information on regression parameters, predicted values, or the coefficient of determination.

Question 20

In a regression problem,if the coefficient of determination is 0.90,this means that:
A) 90% of the y values are positive.
B) 90% of the variation in y can be explained by the regression line
C) 90% of the x values are equal.
D) 90% of the variation in x can be explained by regression line.

Accepted Answer

The coefficient of determination, also known as R-squared, measures the proportion of the variation in the dependent variable (y) that can be explained by the regression line. So, if the coefficient of determination is 0.90, it means that 90% of the variation in y can be explained by the regression line.

Quiz 15: Simple Linear Regression and Correlation

If the standard error of estimate 5_yx = 18 and n = 10, then the error sum of squares, SSE, is:

The following values are listed as coefficients of correlation (r) .The one that indicates an inverse relationship between the two variables x and y is:

For a given value of x,the estimation interval for an individual y observation is called the:

Correlation analysis requires that the scales of measurement be expressed in either:

In simple linear regression,the coefficient of correlation r and the least squares estimate $b _ { 1 }$ of the population slope $\beta _ { 1 }$ :

Which of the following table values would be appropriate for a 95% confidence interval for the mean of y from a simple linear regression problem if the sample size is 7?

Given the least squares regression line $\hat{y}$ = -2.88 + 1.77x,and a coefficient of determination of 0.81,the coefficient of correlation is:

The regression line $\hat{y}$ = 3 + 2x has been fitted to the data points (4,8) , (2,5) , and (1,2) . The residual sum of squares will be:

The number of degrees of freedom associated with the standard error of estimate is

For the values of the coefficient of determination listed below,which one implies the greatest value of SSR (regression sum of squares) given that SST = 500?

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?

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

In publishing the results of some research work,the following values of the coefficient of determination were listed.Which one would appear to be incorrect?

All of the values of an independent variable equal the same number.Regressing a dependent variable on this independent variable will result in a coefficient of determination (r² ) of:

The vertical spread of the data points about the regression line is measured by the:

An indication of no linear relationship between two variables would be:

The value for SSE equals zero.This means that the coefficient of determination (r² ) must equal:

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

Correlation analysis is used to determine the:

In a regression problem,if the coefficient of determination is 0.90,this means that: