Exam 16: A: Simple Linear Regression and Correlation

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 af Twa Highest Paid Perfarmers (4mil) Grass 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 -{Theatre Revenues Narrative} Interpret the value of the slope of the regression line.

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 minimized.

In simple linear regression,most often we perform a two-tail test of the population slope $\beta$ ₁ to determine whether there is sufficient evidence to infer that a linear relationship exists.The null hypothesis is stated as:

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 -{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?

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

Game Show Winnings & 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 -{Game Show Winnings & Education Narrative} Do the tests $\rho$ and $\beta$ ₁ in the previous two questions provide the same results? Explain.

The farther a given value of x is from the mean of x,the ____________________ the estimated error becomes.

U V's and Skin Cancer A medical statistician wanted to examine the relationship between the amount of UV's (x)and incidence of skin cancer (y).As an experiment he found the number of skin cancers detected per 100,000 of population and the average daily sunshine in eight states around the country.These data are shown below. Average Daily UV's 5 7 6 7 8 6 4 3 Skin Cancer per 100,000 7 11 9 12 15 10 7 5 -{Sales and Experience Narrative} Determine the standard error of estimate and describe what this statistic tells you about the regression line.

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. $\begin{array} { | c | c | c | } \hline \text { Textbouk } & \text { Number af Peges } & \text { Selling Price (\$) } \\\hline 1 & 844 & 55 \\2 & 727 & 50 \\3 & 360 & 35 \\4 & 915 & 60 \\5 & 295 & 30 \\6 & 706 & 50 \\7 & 410 & 40 \\8 & 905 & 53 \\9 & 1058 & 65 \\10 & 865 & 54 \\11 & 677 & 42 \\12 & 912 & 58 \\\hline\end{array}$ -{Cost of Textbooks Narrative} Determine the least squares regression line.

The point where confidence intervals and prediction intervals do best is $( \bar { x } , \bar { y } )$ .

The coefficient of determination is equal to the coefficient of correlation squared.

Oil Quality and Price Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel. $\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Covariances Degeres Price Degeres 21.281667 Price 2.026750 0.208933 Analysis of Variance Source DF SS MS F p Regeression 1 2.3162 2.3162 134.24 0.000 Resichul Entar 11 0.1898 0.0173 Total 12 2.5060 -Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y.

A confidence interval estimate for the expected value of y will always be wider than the prediction interval for the same given value of x and the same confidence level.

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

Game Winnings & 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 -{Game Winnings & Education Narrative} Estimate with 95% confidence the average winnings of all contestants who have 15 years of education.

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. AEe 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: -{Oil Quality and Price Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between the quality of oil and price per barrel.

Oil Quality and Price Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel. $\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Covariances Degeres Price Degeres 21.281667 Price 2.026750 0.208933 Analysis of Variance Source DF SS MS F p Regeression 1 2.3162 2.3162 134.24 0.000 Resichul Entar 11 0.1898 0.0173 Total 12 2.5060 -If the value of the sum of squares for error SSE equals zero,then the coefficient of determination must equal zero.

Income and Education A professor of economics wants to study the relationship between income (y in $1000s)and education (x in years).A random sample 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 -{Income and Education Narrative} Estimate the income of an individual with 15 years of education.

The regression line $\hat { y } = 3 + 2 x$ has been fitted to the data points (4,8), (2,5),and (1,2).The sum of the squared residuals will be:

{Sales and Experience Narrative} Estimate with 95% confidence the average monthly sales of all salespersons with 10 years of experience.