Exam 18: Simple Linear Regression and Correlation

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 Predict with 95% confidence the average gross revenue of a movie whose top two stars earn $5.0 million.

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

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. Determine the standard error of the estimate and describe what this statistic sells you about the regression line.

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 Identify possible outliers.

If an estimated regression line has a y-intercept of 10 and a slope of -5, then when x = 0, the estimated value of y is: A. -5 B. 0 C. 5 D. 10

Which of the following best describes if we want to test for a linear relationship between x and y, in regression analysis? \begin{array}{|l|l|}\hline\text { A. } & \text {Conduct a t-test for} \beta_{1} \\\hline \text { B. } & \text { Conduct a t-test for }\rho \\\hline \text { C. } &\text {Conduct a t-test for \beta_{1} or a t-test for \( \beta 0 \). }\\\hline \text { D. } &\text {Conduct a t-test for \( \beta_{1} \) or a t-test for \( \rho \). }\\\hline\end{array}

In a regression problem, if the coefficient of determination is 0.95, this means that: A. 95 \% of the y values are positive. B. 95\% of the variation in y can be explained by the variation in x . C. 95\% of the values are equal. D. 95\% of the variation in x can be explained by the variation in y .

The least squares method requires that the variance $\sigma _ { \varepsilon } ^ { 2 }$ of the error variable $\varepsilon$ is a constant no matter what the value of x is. When this requirement is violated, the condition is called: A. multicollinearity. B. heteroscedasticity. C. homoscedasticity. D. autocorrelation.

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

A regression analysis between y, sales (in $1000) and x, advertising (in $) yielded the least squares line = 60 + 5x. We can interpret the slope by saying that we estimate for each extra $1 spent on advertising that sales will increase by $5 000, on average.

Which of the following statements best describes why a linear regression is also called a least squares regression model? A A linear regression is al so called a least squares regression model because the regression line is calculated by minimizing the square of the difference between each actual x data value and the predicted value. B A linear regression is al so called a least squares regression model because the regression line is calculated by minimizing the sum of the difference between each actual y data value and the predicted y value. C A linear regression is al so called a least squares regression model because the regression line is calculated by minimizing the square of each actual y data value and the predicted y value. D A. linear regression is also called a least squares regression model because the regression line is calculated by minimizing the sum of the square of the differences between each actual y data value and the predicted y value.

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. Coffee sales revenue Temperature 6.50 25 10.00 17 5.50 30 4.50 35 3.50 40 28.00 9 a. Find the least squares regression line. b. Interpret the slope.

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 Identify possible outliers.

The regression line = 2 + 3x has been fitted to the data points (4,11), (2,7), and (1,5). The residual sum of squares will be 10.0.

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 regression equation to determine the predicted values of y.

Predict weekly sales in the fast food restaurant if 5 vouchers are printed in the local newspaper, given, Estimated Sales = 11.5676 + 0.4618.Vouchers Is this a good estimate?

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 Use the regression equation to determine the predicted values of y.

A regression analysis between weight $y$ (in kilograms) and height $x$ (in centimetres) yielded the least squares line = 135 + 6 $x$ . We estimate that for each 1 cm in height, that weight will cedrease by 6 kilograms, on average.

In a simple linear regression problem, the least squares line is = -3.75 + 1.25 $x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be 0.90.

Plot the residuals against the predicted values of y. Does the variance appear to be constant?