Exam 15: Simple Linear Regression and Correlation

Given that $s _ { x } ^ { 2 }$ = 500, $s _ { y } ^ { 2 }$ = 750, $s _ { x y }$ = 100 and n = 6, the standard error of estimate is:

Given that SSE = 150 and SSR = 450, the proportion of the variation in y that is explained by the variation in x is 0.75.

Except for the values r = -1, 0 and 1, we cannot be specific in our interpretation of the coefficient of correlation r. However, when we square it, we produce a more meaningful statistic.

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

The probability distribution of the error variable $\varepsilon$ is supposed to be normal, with mean E( $\varepsilon$ ) = 0 and constant standard deviation $\sigma _ { \varepsilon }$ .

If the standard error of estimate $S _ { ? }$ = 15 and n = 12, then the sum of squares for error, SSE, is:

If cov(x,y) = ?350, $s _ { x } ^ { 2 }$ = 900 and $s _ { y } ^ { 2 }$ = 225, then the coefficient of correlation is:

Which of the following best describes the y-intercept in the simple linear regression model?

Which of the following statements best describes the slope in the simple linear regression model?

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education. The results are as follows. Contestant Years of education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 6 16 300 7 13 650 8 14 400 Calculate the Pearson correlation coefficient. What sign does it have? Why?

The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyse the claim, a university student visits the bookstore and records the number of pages and the selling price of 12 randomly selected books. These data are listed below. Book Number of pages Selling price (\ ) 1 844 55 2 727 50 3 360 35 4 915 60 5 295 30 6 706 50 7 410 40 8 905 53 9 1058 65 10 865 54 11 677 42 12 912 58 Determine the coefficient of determination, and discuss what its value tells you.

The quality of oil is measured in API gravity degrees - the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field, who believes that there is a relationship between quality and price per barrel. Oil degrees API Price per barrel (in \ ) 27.0 12.02 28.5 12.04 30.8 12.32 31.3 12.27 31.9 12.49 34.5 12.70 34.0 12.80 34.7 13.00 37.0 13.00 41.0 13.17 41.0 13.19 38.8 13.22 39.3 13.27 A partial Minitab output follows. Descriptive Statistics Variable Mean StDev SE Mean Degrees 13 34.60 4.613 1.280 Frice 13 12.730 0.457 0.127 Covariances Degrees Price Degrees 21.281667 Price 2.026750 0.208833 Regression Analysis Fredictor Coef StDev Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7% Analysis of Variance Source DF SS MS Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 a. Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate to describe the relationship between the quality of oil and price per barrel. b. Determine the least squares regression line. c. Redraw the scatter diagram and plot the least squares regression line on it. d. Interpret the value of the slope of the regression line.

Regardless of the value of x, the standard deviation of the distribution of y values about the regression line is supposed to be constant. This assumption of equal standard deviations about the regression line is called multicollinearity.

The dean of a faculty of business in Victoria believes that students who do well in 'soft' courses like organisational behaviour do poorly in 'hard' courses like business statistics. In order to test his belief, he takes a random sample of 10 students and records their test grades in organisational behaviour and statistics. The results are shown below. Do these data provide sufficient evidence at the 5% significance level to support the dean's claim? Student Organisational Behaviour Grade Business Statistics Grade 1 C A 2 D A 3 A C 4 B C 5 A D 6 C B 7 B C 8 A C 9 C A 10 B C

When the actual values y of a dependent variable and the corresponding predicted values $\ddot { p }$ are the same, the standard error of estimate, $S _ { \varepsilon }$ , will be 0.0.

The coefficient of determination is the coefficient of correlation squared.

In the first-order linear regression model, the population parameters of the y-intercept and the slope are estimated by:

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education. The results are as follows. Contestant Years of education Winnings Predicted values Residuals 1 11 750 754.2 -4.167 2 15 400 397.5 2.500 3 12 600 665.0 -65.000 4 16 350 308.3 41.667 5 11 800 754.2 45.833 6 16 300 308.3 -8.333 7 13 650 575.8 74.167 8 14 400 486.7 -86.667 Plot the residuals against the predicted values. Does the variance appear to be constant?

A professor of economics wants to study the relationship between income y (in $1000s) and education x (in years). A random sample of eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 Use the predicted and actual values of y to calculate the residuals.

The method of least squares requires that the sum of the squared deviations between actual y values in the scatter diagram and y values predicted by the regression line be minimised.