Exam 16: Simple Linear Regression and Correlation

NARRBEGIN: Trivia Games & Ed.Trivia Games & Education 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 favorite game show. She records their winnings in dollars and the number of 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 0 16 300 7 13 650 8 14 400 NARREND -{Trivia Games & Education Narrative} Interpret the value of the slope of the regression line.

Given the least squares regression line $\hat { y } = 5 - 2 x$ :

NARRBEGIN: Truck Speed & Gas Mileage Truck Speed and Gas Mileage An economist wanted to analyze the relationship between the speed of a truck (x) and its gas mileage (y). As an experiment a truck is operated at several different speeds and for each speed the gas mileage is measured. These data are shown below. Speed 25 35 45 50 60 65 70 Gas Mileage 40 39 37 33 30 27 25 NARREND -{Truck Speed and Gas Mileage Narrative} What does the coefficient of correlation tell you about the direction and strength of the relationship between the two variables?

The standard error of estimate s _{$\varepsilon$} is a measure of the:

If the coefficient of correlation is 0.90, then the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is:

In a simple linear regression model b₀ is the ____________________ of the straight line.

In a simple linear regression model, testing whether the slope $\beta$ ₁ of the population regression line could be zero is the same as testing whether or not the population coefficient of correlation $\rho$ equals zero.

When the variance $\sigma _ { z } ^ { 2 }$ of the error variable $\varepsilon$ is a constant no matter what the value of x is, this condition is called:

NARRBEGIN: Grateful Dead Concert Grateful Dead Concert At a recent Grateful Dead concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year. It is suspected that older concert goers tend to go to more of his concerts in one year than younger concert goers. The data and analysis are shown below. Age 62 57 40 49 67 54 43 65 54 41 Number af 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 An Excel output follows: SUMMARY OUTPUT Regression Statistics Multiple R 0.80203 R Square 0.64326 Adjusted R Square 0.62344 Standard Error 0.93965 Observations 20 $\text { DESCRIPTIVE STATISTICS }$ AGe Concerts Mean 53 Mean 3.65 Standard Error 2.1849 Standard Error 0.3424 Standard Deviation 9.7711 Standard Deviation 1.5313 Sample Variance 95.4737 Sample Variance 2.3447 Count 20 Court 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ ANOVA df SS MS F Signficance F Regression 1 28.65711 28.65711 32.45653 2.1082E-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standard Error tStat p-value Lower 95\% Upper 95\% Intercept -3.01152 1.18802 -2.53491 0.02074 -5.50746 -0.5156 Age 0.12569 0.02206 5.69706 0.00002 0.07934 0.1720 NARREND -{Grateful Dead Concert Narrative} Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a positive linear relationship exists between age and number of concerts attended.

NARRBEGIN: Sales and Experience Sales and Experience The general manager of a chain of hardware stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. Salesperson Years of Exgperience Sales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 6 5 14 7 12 20 8 7 17 9 20 30 10 15 25 NARREND -(Sales and Experience Narrative} Determine the coefficient of determination and discuss what its value tells you about the two variables.

NARRBEGIN: Grateful Dead Concert Grateful Dead Concert At a recent Grateful Dead concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year. It is suspected that older concert goers tend to go to more of his concerts in one year than younger concert goers. The data and analysis are shown below. Age 62 57 40 49 67 54 43 65 54 41 Number af 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 An Excel output follows: SUMMARY OUTPUT Regression Statistics Multiple R 0.80203 R Square 0.64326 Adjusted R Square 0.62344 Standard Error 0.93965 Observations 20 $\text { DESCRIPTIVE STATISTICS }$ AGe Concerts Mean 53 Mean 3.65 Standard Error 2.1849 Standard Error 0.3424 Standard Deviation 9.7711 Standard Deviation 1.5313 Sample Variance 95.4737 Sample Variance 2.3447 Count 20 Court 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ ANOVA df SS MS F Signficance F Regression 1 28.65711 28.65711 32.45653 2.1082E-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standard Error tStat p-value Lower 95\% Upper 95\% Intercept -3.01152 1.18802 -2.53491 0.02074 -5.50746 -0.5156 Age 0.12569 0.02206 5.69706 0.00002 0.07934 0.1720 NARREND -{Grateful Dead Concert Narrative} Determine the standard error of estimate and describe what this statistic tells you about the model's fit.

To create a deterministic model, we start with a probabilistic model that approximates the relationship we want to model.

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

Given that the sum of squares for error is 60 and the sum of squares for regression is 140, then the coefficient of determination is:

If the coefficient of determination is 0.95, this means that 95% of the variation in the independent variable x can be explained by the y variable.

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line: $\hat { y } = 75 + 6 x$ . This implies that if advertising is $800, then the predicted amount of sales (in dollars) is:

NARRBEGIN: Theatre Revenues Theatre Revenues A financier whose specialty is investing in stage productions has observed that, in general, movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid performers in the play for ten recently staged plays. Play Cost of Two Highest Paid Performers ( \mil ) 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 0.7 58 NARREND -{Theatre Revenues Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.

NARRBEGIN: Sunshine and Melanoma Sunshine and Melanoma A medical researcher wanted to examine the relationship between the amount of sunshine (x) in hours, and incidence of melanoma, a type of skin cancer (y). As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Melanoma per 100,000 7 11 9 12 15 10 7 5 NARREND -{Sunshine and Melanoma Narrative} Determine the least squares regression line.

NARRBEGIN: Cost of TextBooks Cost of Textbooks The editor of a higher education book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger textbooks will cost more money. As an experiment to analyze the claim, a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected textbooks. These data are listed below. $\begin{array} { | c | c | c | } \hline \text { Textbook } & \text { Number af Peges } & \text { Selling Price (\$) } \\\hline 1 & 844 & 55 \\2 & 727 & 50 \\3 & 360 & 35 \\4 & 915 & 60 \\5 & 295 & 30 \\6 & 706 & 50 \\7 & 410 & 40 \\8 & 905 & 53 \\9 & 1058 & 65 \\10 & 865 & 54 \\11 & 677 & 42 \\12 & 912 & 58 \\\hline\end{array}$ NARREND -{Cost of Textbooks Narrative} Estimate the selling price for a 650 pages book.

If we are interested in determining whether two variables are linearly related, it is necessary to: