Exam 16: A: Simple Linear Regression and Correlation

When all the actual values of y are equal to their predicted values,the standard error of estimate will be:

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 -The value of the sum of squares for regression SSR can never be larger than the value of sum of squares for error 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} Predict with 95% confidence the winnings of a contestant who has 10 years of education.

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 -In a simple linear regression problem,the least squares line is $\hat { y } = - 3.75 + 1.25 x$ ,and the coefficient of determination is 0.81.The coefficient of correlation must be -0.90.

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} Can we conclude at the 1% significance level that there is a linear relationship between sunshine and skin cancer?

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:

Given the least squares regression line $\hat { y } = - 2.48 + 1.63 x$ ,and a coefficient of determination of 0.81,the coefficient of correlation is:

A prediction interval for a particular y is always ____________________ than a confidence interval for the mean of y.

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 -The value of the sum of squares for regression SSR can never be larger than the value of total sum of squares SST.

A store manager gives a pre-employment examination to new employees.The test is scored from 1 to 100.He has data on their sales at the end of one year measured in dollars.He wants to know if there is any linear relationship between pre-employment examination score and sales.An appropriate test to use is the t-test of the population correlation coefficient.

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} Are the two highest paid performers worth all the money paid for them? Comment using the statistical analyses you have done.

A regression analysis between weight (y in pounds)and height (x in inches)resulted in the following least squares line: $\hat { y } = 135 + 6 x$ .This implies that if the height is increased by 1 inch,the weight is expected to increase by an average of 6 pounds.

In a simple linear regression problem,r and b₀:

Allman Brothers Concert At a recent Allman Brothers 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.The following data were collected: 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 af Concerts 3 2 4 5 4 5 4 2 1 3 An Excel output follows: -{Allman Brothers Concert Narrative} Determine the least squares regression line.

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} Determine the least squares regression line.

A straight line regression model with only one independent variable is called a(n)____________________-order linear model.

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

The standard error of the estimate is a measure of the:

If the regression line is horizontal,then we conclude that y ____________________ (is/is not)related to x.

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 10 years of education.