Exam 16: A: Simple Linear Regression and Correlation

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

Given that $s _ { x } ^ { 2 } = 500 , s _ { y } ^ { 2 } = 750 , \operatorname { cov } ( x , y ) = 100$ and n = 6,the standard error of estimate is:

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 -When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same,the standard error of estimate s _{$\varepsilon$} will be 0.0.

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.

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} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate.

If the sum of squared residuals is zero,then the:

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} Determine the standard error of estimate and describe what this statistic tells you.

In the simple linear regression model,the y-intercept represents the:

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: -{Grateful Dead Concert Narrative} Do the $\rho$ and $\beta$ ₁ tests in the previous two questions provide the same results? Explain.

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} Do the tests of $\rho$ and $\beta$ ₁ in the previous two questions provide the same results? Explain.

Rock Concert Revenues A financier whose specialty is investing in rock concerts has observed that,in general,concerts with "big-name" stars seem to generate more revenue than those concerts 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 concert for ten concert tours. Concert Cost of Twa Highest Paid Perfarmers ( \mil ) 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 -{Rock Concert Revenues Narrative} Calculate the Pearson correlation coefficient.Interpret the results.

In the simple linear regression model,the slope represents the:

The deviations between the actual data points and the fitted values from the model are called ____________________.

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 -{UV's and Skin Cancer Narrative} Calculate the standard error of estimate,and describe what this statistic tells you about the regression line.

In regression analysis,the residuals represent the:

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} Conduct a test of the population slope to determine at the 5% significance level whether a negative linear relationship exists between years of education and TV game shows' winnings.

SSE stands for ____________________ of squares for ____________________.

Movie Revenues 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 $ millions)given to the two highest-paid performers in the movie for ten recently released movies. Movie Cost of Twa Highest Paid Perfarmers ( \mil ) 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 -{Movie Revenues Narrative} Estimate with 95% confidence the average gross revenue of a movie whose top two stars earn $5.0 million.

In the first order linear regression model,the population parameters of the y-intercept and the slope are estimated,respectively,by:

Movie Revenues 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 $ millions)given to the two highest-paid performers in the movie for ten recently released movies. Movie Cost of Twa Highest Paid Perfarmers ( \mil ) 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 -{Cost of Books Narrative} Predict with 90% confidence the selling price of a book with 900 pages.