Exam 16: Simple Linear Regression and Correlation

We check for normality by drawing a(n) ____________________ of the residuals.

NARRBEGIN: Game Show Winnings & Ed.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 NARREND -{Game Show Winnings & Education Narrative} Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a negative linear relationship exists between years of education and TV game shows' winnings.

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

There is ____________________ error in estimating a mean than in predicting an individual value.

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} Compute the standardized residuals.

Which of the following assumptions concerning the probability distribution of the random error term is stated incorrectly?

NARRBEGIN: Sunshine and Melanoma Sunshine and Melanoma A medical researcher wanted to examine the relationship between the amount of sunshine (x) in hours, and incidence of melanoma, a type of skin cancer (y). As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Melanoma per 100,000 7 11 9 12 15 10 7 5 NARREND -{Sunshine and Melanoma Narrative} Calculate the residual corresponding to the pair (x, y) = (8, 15).

The spread in the residuals should increase as the predicted value of y increases.

If you take a particular x value and plug it into a regression line equation, the result is a(n) ____________________ estimate for y.

Graphically, a prediction interval is represented as two ____________________ lines.

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} Estimate with 95% confidence the average winnings of all contestants who have 15 years of education.

NARRBEGIN: Sunshine and Melanoma Sunshine and Melanoma A medical researcher wanted to examine the relationship between the amount of sunshine (x) in hours, and incidence of melanoma, a type of skin cancer (y). As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Melanoma per 100,000 7 11 9 12 15 10 7 5 NARREND -{Sunshine and Melanoma Narrative} Estimate the number of skin cancer cases per 100,000 people who live in a state that gets 6 hours of sunshine on average.

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

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: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0208837 Rederatian Antalyis predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S=0.1314R-Sq=92.46\%R-Sq(ad)=91.7\% Analysis of Variance Source DF SS MS F P Regeression 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} Use the predicted values and the actual values of y to calculate the residuals.

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: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0208837 Rederatian Antalyis predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S=0.1314R-Sq=92.46\%R-Sq(ad)=91.7\% Analysis of Variance Source DF SS MS F P Regeression 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} Use the regression equation $\tilde { y } = 9.4349 + 0.095235 x$ to determine the predicted values of y.

We check for normality by drawing a pie chart of the residuals.

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.

In regression analysis, you predict the value of one variable on the basis of one or more other related variables. The variable being predicted is called the ____________________ variable, and the related variables used to make the prediction are called ____________________ variables.

NARRBEGIN: Speed vs Gas Mileage Speed vs Gas Mileage An economist wanted to analyze the relationship between the speed of a car (x) and its gas mileage (y). As an experiment a car 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 -{Car Speed and Gas Mileage Narrative} Determine the least squares regression line.

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} Predict with 99% confidence the gas mileage of a car traveling 55 mph.