Exam 15: Simple Linear Regression and Correlation

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 Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between TV game show contestants' years of education and their winnings.

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. Age 62 57 40 49 67 54 43 65 54 41 Number of concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number of Concerts 3 2 4 5 4 5 4 2 1 3 a. Calculate the Pearson correlation coefficient. What sign does it have? Why? b. Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between age and number of concerts attended.

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 Calculate and interpret a 95% confidence interval for the population slope.

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

One method of diagnosing autocorrelation is to plot the residuals against the time periods to see whether some pattern emerges.

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%.

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.

In a simple linear regression problem, the following statistics are calculated from a sample of 10 observations: $\sum ( x - \bar { x } ) ( y - \bar { y } )$ = 2250, $S _ { x }$ = 10, $\sum x$ = 50, $\sum y$ = 75 The least squares estimates of the slope and y-intercept are respectively:

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 Predict with 95% confidence the income of an individual with 10 years of education.

A regression analysis between y, sales (in $1000) and x, advertising (in $) yielded the least squares line $\ddot { p }$ = 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.

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

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 a. Write down the regression equation b. Interpret the slope.

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 Use the regression equation to determine the predicted sales for number of vouchers in the sample, given, Estimated Sales = 11.5676 + 0.4618.Vouchers

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 Find the least squares regression line.

Which of the following best describes the coefficient of determination?

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 SUMMARY OUTPUT RegressionStatistios Multiple R 0.8644 RSquare 0.7472 Adjusted RSquare 0.6840 Standard Error 5.2027 Observations 6 ANOVA df SS MS F Significance F Regression 1 320.0617 320.0617 11.8244 0.0263 Residual 4 108.2716 27.0679 Total 5 4283333 Coefficient StandardError tStat P-value Lover 95\% Upper 95\% Intercept 27.7179 5.6629 4.8946 0.0081 11.9952 43.4406 Temperature -0.6943 0.2019 -3.4387 0.0263 -1.2549 -0.1337 (a) Estimate daily hot coffee sales revenue on a day of 38 degrees Celsius. (b) Is your prediction in part (a) reasonable?

Given that cov(x,y) = 8, $s _ { y } ^ { 2 }$ = 14, $S _ { x } ^ { 2 }$ = 10 and n = 6, the value of the sum of squares for error, SSE, is 38.

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 Predict with 95% confidence the winnings of a contestant who has 15 years of education.

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 Estimate with 90% confidence the mean selling price of all books with 900 pages.