Question 1

If there is a negative correlation between the independent variable x and the dependent variable y, then to test this, the appropriate null and alternative hypotheses would be  .

Accepted Answer

False

Question 2

In regression analysis, a careful study of the differences,  , between observed and estimated y values, given x (in order to decide whether crucial assumptions are fulfilled that allow valid inferences about the true regression line to be made from an estimated regression line) is called residual analysis.

Accepted Answer

Residual analysis is a careful study of the differences between observed and estimated y values in regression analysis. It helps in checking whether the assumptions necessary to make valid inferences about the true regression line from an estimated regression line are fulfilled.

Question 3

Large coefficient of determination value will result in a small standard error of the estimate for the regression model, thus providing prediction intervals that are narrow.

Accepted Answer

A large coefficient of determination (R²) indicates a good fit of the model to the data, but it does not directly determine the size of the standard error of the estimate or the width of prediction intervals, which also depend on other factors like sample size and variability of the data.

Question 4

In developing a 90% confidence interval for the average value of y from a simple linear regression problem involving 12 observations, the appropriate table value would be 1.796.

Accepted Answer

For a 90% confidence interval in a simple linear regression with 12 observations (n=12), the degrees of freedom for the t-distribution would be n-2 = 10. The appropriate t-value should be looked up in a t-table or calculated using statistical software for 10 degrees of freedom, not directly stated as 1.796.

Question 5

If the correlation coefficient for two variables is found to be .094, then the scatterplot will show the data upward sloping from lower left to upper right.

Accepted Answer

A positive correlation coefficient (between 0 and 1) indicates a positive relationship between the variables, meaning that as one variable increases, the other variable tends to increase as well. The upward sloping pattern from lower left to upper right on a scatterplot is often indicative of a positive correlation.

Question 6

The regression model  = 36.5 + 20.1x has been computed based on a sample of 50 observations. One observation in the sample was (x, y) = (14, 350.9). Given this, the residual value for this observation is 33.

Accepted Answer

The answer of The regression model  = 36.5 +...

Question 7

In simple linear regression, one can use the plot of residuals versus the fitted values of y to check for a constant variance as well as to make sure that the linear model is in fact adequate.

Accepted Answer

This is a true statement. The plot of residuals versus the fitted values is known as a "residual plot" and can be used to check for violations of the assumptions of linear regression, including the constant variance assumption and the linearity assumption. If the plot shows a clear pattern (such as a funnel shape), it may indicate that the assumptions are being violated and the linear model is not appropriate.

Question 8

If the correlation coefficient between two variables is very close to zero, this means that there is no relationship between the two variables.

Accepted Answer

A correlation coefficient close to zero suggests that there is a weak linear relationship between the variables, but it does not rule out the possibility of a strong non-linear relationship.

Question 9

Given that n = 37, and the value of sample Spearman rank correlation coefficient  = 0.35, the value of the test statistic for testing  .

Accepted Answer

The test statistic for testing the significance of the Spearman rank correlation coefficient is calculated as follows:

t = r √((n-2)/(1-r^2))

where r is the sample Spearman rank correlation coefficient and n is the sample size.

Substituting the given values, we get:

t = 0.35 √((37-2)/(1-0.35^2)) = 2.38

This value can be compared with a critical value from the t-distribution with n-2 degrees of freedom to determine if the correlation coefficient is statistically significant. However, the question does not provide the critical value or the significance level, so we cannot determine if the test statistic is significant or not. Therefore, the answer is A) True, as we can calculate the test statistic without additional information.

Question 10

The normal probability plot is a graph that plots the residuals against the expected value of that residual if it had come from a normal distribution. When the residuals are normally distributed or approximately so, the plot should appear as a straight line, sloping upward at a 45° angle.

Accepted Answer

This statement accurately describes the characteristics and interpretation of a normal probability plot.

Question 11

Simple regression analysis is a statistical technique that establishes an index that provides, in a single number, a measure of the strength of association between two variables.

Accepted Answer

This statement is true. Simple regression analysis is a statistical technique that measures the relationship between two variables, typically by establishing a linear equation that predicts one variable based on the other variable. The resulting index, expressed as a correlation coefficient or coefficient of determination, provides a measure of the strength and direction of the association between the two variables.

Question 12

In simple linear regression analysis, if the correlation coefficient between the independent variable x and the dependent variable y is -.85, this means that the scatterplot generated by the same data values would show points that would fall on a straight line with slope equal to -.85.

Accepted Answer

The correlation coefficient between x and y being -.85 indicates a strong negative linear association between the two variables, but it does not determine the exact slope of the line. Other factors such as the range and variability of the data can also affect the slope of the regression line.

Question 13

A perfect correlation between two variables will always produce a correlation coefficient of + 1.0.

Accepted Answer

A perfect correlation between two variables can produce a correlation coefficient of +1.0 for a perfect positive correlation or -1.0 for a perfect negative correlation.

Question 14

In a simple linear regression problem, the least squares line is  = 2.73 - 1.02x, and the coefficient of determination is 0.7744. The correlation coefficient must be -0.88.

Accepted Answer

The correlation coefficient (r) is equal to the square root of the coefficient of determination (R-squared). Therefore, if R-squared is 0.7744, then r is the square root of 0.7744, which is approximately -0.88. Therefore, the answer is A) True.

Question 15

The prediction interval developed from a simple linear regression model will be very narrow when the value of x used to predict y is equal to the mean value  .

Accepted Answer

When the value of x used to predict y is equal to the mean value, the prediction interval will be very narrow because the predicted value of y will be very close to the mean value of y.

Question 16

When regression analysis is used for prediction, the confidence interval for the average y given x will be wider than the prediction interval for a particular value of y given x.

Accepted Answer

The prediction interval for a particular value of y given x is wider than the confidence interval for the average y given x because it accounts for the additional uncertainty in predicting a single outcome rather than the mean outcome.

Question 17

In regression analysis, a graph of each residual against the corresponding fitted value is called a scatter diagram.

Accepted Answer

In regression analysis, a graph of each residual against the corresponding fitted value is called a residual plot, not a scatter diagram. A scatter diagram typically refers to a graph of two variables against each other to explore their relationship.

Question 18

In a simple linear regression analysis, it was stated that the correlation between starting salary and years of experience is 0.80. This indicates that 80% of the variation in starting salary is explained by years of experience.

Accepted Answer

Correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1. A correlation of 0.80 indicates a strong positive linear relationship between starting salary and years of experience. However, it does not mean that 80% of the variation in starting salary is explained by years of experience. The coefficient of determination (R-squared) measures the proportion of variability in the dependent variable (i.e., starting salary) that is explained by the independent variable(s) (i.e., years of experience) in a linear regression model. The R-squared value can range from 0 to 1, where 0 indicates no linear relationship and 1 indicates a perfect linear relationship. Therefore, we cannot conclude anything about the proportion of variation explained from the correlation coefficient alone.

Question 19

If all the points in a scatterplot lie on the least squares regression line, then the correlation coefficient must be 1.0

Accepted Answer

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

Question 20

In simple linear regression analysis, if the independent variable x and the dependent variable y are highly correlated, this does not only mean that they are linearly related, but it also means that a change in x will cause a change in y.

Accepted Answer

High correlation between x and y in simple linear regression analysis indicates a strong linear relationship, but it does not imply causation. Correlation does not establish that a change in x will cause a change in y.

Quiz 12: Linear Regression and Correlation

If there is a negative correlation between the independent variable x and the dependent variable y, then to test this, the appropriate null and alternative hypotheses would be .