Exam 12: Linear Regression and Correlation

The residuals are observations of the error variable . Consequently, the minimized sum of squared deviations is called the sum of squares for error, denoted SSE.

One way to measure the strength of the relationship between the response variable y and the predictor variable x is to calculate the coefficient of determination; that is, the proportion of the total variation in y that is explained by the linear regression of y on x.

In simple linear regression, if the estimated values and the corresponding actual values are equal, then the standard error of estimate, SE( ), must equal -1.0.

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 .

If the coefficient of determination value for a simple linear regression model is .90, then the correlation coefficient between the two variables will be .81.

In regression analysis, the independent variable is a variable whose value is known and is being used to explain or predict the value of another variable.

Which of the following statements is false regarding the residuals in simple linear regression model?

Given the least squares regression line = 5 - 2x:

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.

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

The sign of the correlation coefficient in a simple linear regression model will always be the same as the sign of the y-intercept coefficient .

In simple linear regression, the plot of residuals versus fitted values should:

Evidence supports using a simple linear regression model to estimate a person's weight based on a person's height. Let x be a person's height (measured in inches) and y be the person's weight (measured in pounds). A random sample of eleven people was selected and the following data recorded: The following output was generated for the data: Based on the scatterplot above, does a simple linear regression model seem appropriate? ______________ Justify your answer. ________________________________________________________ Use the printout to find the least-squares prediction line. = ______________ Based on the printout, do there appear to be any outliers in the data? ______________ Justify your answer. ________________________________________________________ Consider the following residual plot of the residuals versus the fitted values. What conclusion, if any, can be drawn from the plot? ________________________________________________________ Consider the following normal probability plot of the residuals. What conclusion can be drawn from the plot? ________________________________________________________ Based on the previous two plots, should you use the model in the computer printout to predict weight? ______________ Justify your answer. ________________________________________________________

A professor of economics wants to study the relationship between income (y in $1000s) and education (x in years). A random sample eight individuals is taken and the results are shown below. Determine the standard error of estimate. s_r = ______________ Describe what this statistic tells you about the regression line. ____________________________ Determine the coefficient of determination. R² = ______________ Discuss what its value tells you about the two variables. ________________________________________________________ Calculate the Pearson correlation coefficient. r = ______________ Why does it have the sign it has? ________________________________________________________ Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between years of education and income. Test statistic: t = ______________ Rejection Region: Reject H₀ if | t | > ______________ Conclusion: ______________ A linear relationship ______________ between years of education and income.

In a simple linear regression problem, if the coefficient of determination is 0.96, this means that:

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

The coefficient of determination measures the amount of:

In a simple linear regression problem including n = 10 observations, which of the following table values would be appropriate for a 95% confidence interval estimation for the average value of y?

In a simple linear regression problem, the following statistics are calculated from a sample of 10 observations: . The least squares estimates of the slope and y-intercept are respectively:

When the actual values y of a dependent variable and the corresponding predicted values are the same, the standard error of estimate will be -1.0.