Exam 16: Simple Linear Regression and Correlation

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} Interpret the value of the slope of the regression line.

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

The value of the sum of squares for regression SSR can never be smaller than 0.0.

NARRBEGIN: Accidents & Rain 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 NARREND -{Accidents and Rain Narrative} Estimate the number of accidents in a day with 0.25 inches of rain.

NARRBEGIN: Movie Revenues 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 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 -{Movie Revenues Narrative} Predict with 95% confidence the gross revenue of a movie whose top two stars earn $5.0 million.

NARRBEGIN: Game Winnings & Ed.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 NARREND -{Game Winnings & Education Narrative} Use the predicted and actual values of y to calculate the residuals.

For a regression analysis to be valid, the error variable must have a standard deviation that is ____________________ regardless of the value of x.

If the coefficient of determination is 1.0, then the coefficient of correlation must be 1.0.

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

The variance of the error variable $\sigma _ { z } ^ { 2 }$ is required to be constant. When this requirement is satisfied, the condition is called homoscedasticity.

NARRBEGIN: UV's and Skin Cancer 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 NARREND -{UV's and Skin Cancer Narrative} Calculate the coefficient of determination and interpret it.

NARRBEGIN: Telemarketing Sales and E Telemarketing Sales and Experience The general manager of a telemarketing company believes that experience is the most important factor in determining the level of success of a telemarketer. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected telemarketers. These data are listed below. Telemarketer Years of Experience 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 -{Telemarketer Sales and Experience Narrative} Plot the residuals against the predicted values of y. What does the graph tell you?

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

In a simple linear regression problem, r and b₀:

In a simple linear regression problem, the following statistics are calculated from a sample of 10 observations: $\sum ( x - \bar { x } ) ( y - \bar { y } ) = 2250 , s _ { x } = 10 , \sum x = 50 , \sum y = 75$ . The least squares estimates of the slope and y-intercept are, respectively,

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} Calculate the coefficient of determination and interpret its value.

NARRBEGIN: UV's and Skin Cancer 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 NARREND -{UV's and Skin Cancer Narrative} What does the coefficient of correlation calculated tell you about the direction and strength of the relationship between the two variables?

NARRBEGIN: Sales and Experience Sales and Experience The general manager of a chain of department 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. Sales person Years of Experience 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} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.

If the standard error of estimate is zero, then:

NARRBEGIN: Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oil degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Descriptive Statistics Variable Mean StDev SE Mean Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0.208833 $\text { Regression Analysis }$ Predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 $\mathrm{S}=0.1314 \quad \mathrm{R}-\mathrm{Sq}=92.46 \% \quad \mathrm{R}-\mathrm{Sg}(\mathrm{adj})=91.7 \%$ $\text { Analysis of Variance }$ Source DF SS MS F P Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 NARREND -{Oil Quality and Price Narrative} Determine the least squares regression line.