Exam 15: Simple Linear Regression and Correlation

The least squares method for determining the best fit minimises:

Predict weekly sales in the fast food restaurant if 10 vouchers are printed in the local newspaper, given Estimated Sales = 11.5676 + 0.4618. Vouchers and R² = 0.7267. Is this a good estimate?

When the sample size n is greater than 30, the Spearman rank correlation coefficient $r _ { s }$ is approximately normally distributed with:

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: $\hat { y }$ = 1.85 + 2.40x. Model 2: $\hat { y }$ = 1.79 + 2.54x. Using the least squares method, which of these regression models provides the better fit to the data? Why?

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.

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 gross revenue of a movie whose top two stars earn $5.0 million.

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be +1.0.

The standard error of estimate, $S _ { ? }$ , is given by:

A regression line using 25 observations produced SSR = 118.68 and SSE = 56.32. The standard error of estimate was:

Which of the following best describes the value of the slope, if the coefficient of determination is 0.95?

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 predicted and actual values of y to calculate the residuals.

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 SUMMARY OUTPUT Regression Statistics Multiple R 0.8524 R Square 0.7267 Adjusted R Square 0.6925 Standard Error 1.4301 Observations 10 ANOvA df SS MS F Significance F Regression 1 43.4982 43.4982 21.2682 0.0017 Residual 8 16.3618 2.0452 Total 9 59.8600 Coeficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 11.5676 0.8341 13.8679 0.0000 9.6441 13.4912 Number of vouchers 0.4618 0.1001 4.6117 0.0017 0.2309 0.6927 Interpret the coefficient of determination.

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 Compute the standardised residuals.

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 Calculate the Pearson correlation coefficient. What sign does it have? Why?

A direct relationship between an independent variable x and a dependent variably y means that the variables x and y increase or decrease together.

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 Number ofvouchers Sales 4 12.8 Mean 7 Mean 14.8 7 15.4 Standard Error 1.5055 Standard Error 0.8155 5 13.9 Median 5.5 Median 14.65 3 11.2 Mode 5 Mode 13.9 19 18.7 Standard Deviation 4.7610 Standard Deviation 2.5790 10 17.9 Sample Variance 22.6667 Sample Variance 6.6511 8 16.8 Kurtosis 4.7702 Kurtosis -1.1563 6 15.9 Skewness 2.0386 Skewness 0.0535 3 11.5 Rarge 16 Range 7.5 5 13.9 Minimum 3 Minimum 11.2 Maximum 19 Maximum 187 Sum 70 Sum 148 Count 10 Count 10 SUMMARY OUTPUT RegressionStatistics Multiple R 0.8524 RSquare 0.7267 Adjusted R Square 0.6925 Standard Error 1.4301 Observations 10 ANOvA df SS MS F significance F Regression 1 43.4982 43.4982 21.2682 0.0017 Residual 8 16.3618 2.0452 Total 9 59.8600 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 11.5676 0.8341 13.6579 0.0000 9.6441 13.4912 Number of vouchers 0.4618 0.1001 4.6117 0.0017 0.2309 0.6927 Determine the standard error of the estimate and describe what this statistic sells you about the regression line.

A consultant for a beer company wanted to determine whether those who drink a lot of beer actually enjoy the taste more than those who drink moderately or rarely. She took a random sample of eight men and asked each how many beers they typically drink per week. She also asked them to rate their favourite brand of beer on a 10-point scale (1 = bad, 10 = excellent). The results are shown below. Can we infer at the 5% significance level that frequent beer drinkers rate their favourite beer more highly than less frequent drinkers? Beer drinker Typical weekly consumption Rating 1 4 6 2 3 6 3 12 9 4 15 8 5 7 8 6 9 6 7 1 5 8 10 8

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

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 Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between payment to the two highest-paid performers and gross revenue.

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 Identify possible outliers.