Exam 18: Simple Linear Regression and Correlation

The residuals are observations of the error variable $\varepsilon$ . Consequently, the minimised square of deviations is called the sum of squares for error, denoted SSE.

The manager of a fast food restaurant wants to determine how sales in a given week are related to the number of discount vouchers (#) printed in the local newspaper during the week. The number of vouchers and sales ($000s) from 10 randomly selected weeks is given below with Excel regression output. Number of vouchers Sales 4 12.8 7 15.4 5 13.9 3 11.2 19 18.7 10 17.9 8 16.8 6 15.9 3 11.5 5 13.9 Identify possible outliers.

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 Determine the coefficient of determination and discuss what its value tells you about the two variables.

Pop-up coffee vendors have been popular in the city of Adelaide in 2013. A vendor is interested in knowing how temperature (in degrees Celsius) impacts daily hot coffee sales revenue (in $00's). A random sample of 6 days was taken, with the daily hot coffee sales revenue and the corresponding temperature of that day noted. Excel regression output given below. Coffee sales revenue Temperature 6.50 25 10.00 17 5.50 30 4.50 35 3.50 40 28.00 9 Test the significance of the slope, against a two-tailed alternative, at the 5% level of significance.

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 Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate.

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 Determine the least squares regression line.

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.

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line y-hat = 77 +8x. This implies that if advertising is $600, then the predicted amount of sales (in dollars) is $4877.

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.

A statistician investigating the relationship between the amount of precipitation (in inches) and the number of car accidents gathered data for 10 randomly selected days. The results are presented below. Day Precipitation Number of accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 8 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 Calculate the Spearman rank correlation coefficient, and test to determine at the 5% significance level whether we can infer that a linear relationship exists between the number of accidents and the amount of precipitation.

Given that cov(x,y) = 8.5, $s _ { y } ^ { 2 }$ = 8 and $S _ { x } ^ { 2 }$ = 10, the value of the coefficient of determination is 0.95.

In testing the hypotheses: $H _ { 0 } : \rho _ { s } = 0$ $H _ { 1 } : \rho _ { s } \neq 0$ , 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

A scatter diagram includes the following data points: x 3 2 5 4 5 y 9 6 11 11 15 Two regression models are proposed: Model 1: y-hat = 1.85 + 2.40x. Model 2: y-hat = 1.79 + 2.54x. Using the least squares method, which of these regression models provides the better fit to the data? Why?

Given that the sum of squares for error is 50 and the sum of squares for regression is 140, the coefficient of determination is: A. 0.736. B. 0.357. C. 0.263. D. 2.800.

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.

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.

Given the least squares regression line y-hat = 3.52 - 1.27x, and a coefficient of determination of 0.81, the coefficient of correlation is: A. -0.85 B. 0.85 C. -0.90 D. 0.90

A financier whose specialty is investing in movie productions has observed that, in general, movies with 'big-name' stars seem to generate more revenue than those movies whose stars are less well known. To examine his belief, he records the gross revenue and the payment (in $ million) given to the two highest-paid performers in the movie for 10 recently released movies. Movie Cost of two highest- paid performers (\ ) Gross revenue (\ ) 1 5.3 48 2 7.2 65 3 1.3 18 4 1.8 20 5 3.5 31 6 2.6 26 7 8.0 73 8 2.4 23 9 4.5 39 10 6.7 58 Use the predicted and actual values of y to calculate the residuals.

The Pearson coefficient of correlation r equals 1 when there is/are no: A. explained variation B. unexplained variation C. y -intercept in the mode D. outliers.

If the coefficient of correlation is -0.7, then the percentage of the variation in y that is explained by the regression line is 70%.