Exam 16: A: Simple Linear Regression and Correlation

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

The confidence interval estimate of the expected value of y for a given value x,compared to the prediction interval of y for the same given value of x and confidence level,will be:

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} Conduct a test of the population slope to determine at the 5% significance level whether a positive linear relationship exists between years of experience and sales.

Wayne Newton Concert At a recent Wayne Newton 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: -{Wayne Newton Concert Narrative} Predict with 95% confidence the number of concerts attended by a 45 year-old individual.

If the coefficient of determination is 0.95,this means that 95% of the y values were predicted correctly by the regression line.

The regression line $\hat { y } = 2 + 3 x$ has been fitted to the data points (4,11), (2,7),and (1,5).The sum of squares for error will be 10.0.

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} Determine the standard error of estimate and describe what this statistic tells you about the model's fit.

The following 10 observations of variables x and y were collected. 1 2 3 4 5 6 7 8 9 10 y 25 22 21 19 14 15 12 10 6 2 Find the least squares regression line,and the estimated value of y when x = 3.

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

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

For a regression analysis to be valid,the error variable must have a(n)____________________ distribution.

Cost of Books 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 analyze the claim,a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected books.These data are listed below. $\begin{array} { | c | c | c | } \hline \text { Bank } & \text { Number af Peges } & \text { Selling Price (\$) } \\\hline 1 & 844 & 55 \\2 & 727 & 50 \\3 & 360 & 35 \\4 & 915 & 60 \\5 & 295 & 30 \\6 & 706 & 50 \\7 & 410 & 40 \\8 & 905 & 53 \\9 & 1058 & 65 \\10 & 865 & 54 \\11 & 677 & 42 \\12 & 912 & 58 \\\hline\end{array}$ -{Cost of Books Narrative} Determine the coefficient of determination and discuss what its value tells you.

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

Accidents and Rain A statistician investigating the relationship between the amount of rain (in inches)and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year.The results are shown below. Day Rain Number of Accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 -{Accidents and Rain Narrative} What does the slope of the least squares regression line tell you?

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} Does this data provide sufficient evidence at the 5% significance level to infer that a linear relationship exists between speed and gas mileage?

If the standard error of estimate s _{$\varepsilon$} = 20 and n = 10,then the sum of squares for error,SSE,is:

A regression analysis between sales (in $1000)and advertising (in $100)resulted in the following least squares line: $\hat { y } = 77 + 8 x$ .This implies that if advertising is $600,then the predicted amount of sales (in dollars)is $125,000.

A zero correlation coefficient between a pair of random variables means that there is no linear relationship between the random variables.

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 simple linear regression,the denominator of the standard error of estimate s _{$\varepsilon$} is $\sqrt { n - 2 }$ .

For a regression analysis to be valid,the value of the error variable associated with any particular value of y is ____________________ of the value of the error variable associated with any other value of y.