d) What is the slope? Interpret in the context of the problem.

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1 Stat and Data Analysis 4.2 More LSRL 1. Open the group USEDCAR to get the lists USEDA and USEDB. USEDA is the age of the used Corolla. USEDB is the price listed in the local paper for the car. a) Create the scatterplot. Describe the association. Negative Linear Strong(r = ) No outliers b) Write the LSRL. y = -924x c) What is the correlation? What does it tell you? r = ; Strong, negative linear association d) What is the slope? Interpret in the context of the problem. Slope = -924; For every increase of 1 year in the age of the car there tends to be a decrease of 924 dollars in the selling price. e) If you want to sell a 7-year-old Corolla, what price seems appropriate? y = -924(7) y = $

2 f) What is the residual for both cars that were 3 years old? y = -924(3) y = $ e = e = $ g) You have the chance to buy one of two cars. They are about the same age and appear to be in equally good condition. Would you rather buy the one with a positive residual or a negative residual? You want the one with the negative residual because the actual price is less than you would expect. So it s cheaper. h) Would this regression model be useful in establishing a fair price for a 20-yearold car? No. 20 years is too far out of the range of the data. i) What is R 2? What does this mean in the context of the problem? R 2 = 89.4% 89.4% of the changes in selling price can be explained by changes in the age of the car. j) What percent of the variability in price is not being explained by the linear model? 10.6% of the variability in selling price is due to other factors. k) What other factors may be contributing to the change in price? Condition of the car. Model Etc. l) Create and sketch the residual plot m) Does the linear model seem an appropriate one? There is no obvious curve in the residual plot so the linear model is appropriate.

3 2. The following table pertains to basketball players selected in the first round of the 1991 NBA draft. It lists the draft number of each player(pick) and the annual salary(salary) (in $) of the contract that the player signed. The two missing entries are for players who signed with European teams. a) Create and describe a scatterplot and describe the plot. Negative Linear Strong(r = ) No outliers b) Compute the LSRL, and r and r 2. Add the line to your plot. y = x r = R 2 = 78.7% c) Interpret the slope of the LSRL in the context of the problem. For every increase of 1 in the pick number from the draft there tends to be a decrease of 98, dollars in the salary. d) What is R 2? What does this mean in the context of the problem? 78.7% of the changes in the salary can be explained by changes in the draft number. e) Calculate the predicted salary for a player picked 12 th. Also calculate the residual for this value. Was your prediction an overestimate or an underestimate? y = (12) e = y = $ e = -$ Overestimate

4 f) What salary would the line predict for the players picked 15 th and 25 th? y = (15) y = (25) y = $ y = $ g) Create and sketch a residual plot. h) Does a linear model appear to be an appropriate model? Why or why not? The linear model does not seem to appropriate because there is a curve in the residual plot. i) For observations with positive residual values, is the actual salary greater or less than the predicted salary? Are these over- or underestimates? Actual is greater than predicted. Underestimates j) For observations with negative residual values, is the actual salary greater or less than the predicted salary? Are these over- or underestimates? Actual is less than predicted. Overestimates k) Do there appear to be any outliers? No.

5 3. You have lists MILE and TOLL. This is the mileage traveled by drivers on the PA turnpike and the Toll that they paid (in dollars). a) Create and sketch a scatterplot of Mile (x) versus Toll (y). Describe the plot. Positive Linear Strong (r = 0.999) No outliers b) Calculate the LSRL, r and r 2. Add the line to the plot. y = 0.040x r = R 2 = 99.8% c) Interpret the slope in the context of the problem. For every increase of 1 mile in the distance there tends to be an increase of 0.04 dollars in the toll. d) Interpret R 2 in the context of the problem. 99.8% of the changes in tolls can be explained by changes in the miles. e) Predict the Toll for an exit at mile How confident are you in this prediction? y = 0.040(109.9) y = $ is within the range of the data and R 2 = 99.8% so I am very confident in this prediction. f) Find the residual for the toll for mile y = 0.040(201.3) y = $7.936 e = e = $0.264 g) Do you think line does a good job of describing our data? From the scatterplot the line looks like it is fitting the data pretty well.

6 h) Create and sketch the residual plot. i) Using the residual plot, is the line an appropriate model for our data? Why? The residual plot shows an apparent curve so the linear model is not an appropriate model. j) But (based on our original scatterplot) is our line a GOOD model for our data? Since the scatterplot looked linear and correlation and R 2 were so strong we still think it s a good model.

7 4. The list LFEXP provides information on the life expectancies for a sample of 22 countries. The list PERTV gives the average number of people per television set in each country. So a value of 1 would mean that on average, there is 1 person for every TV in the country. A value of 50 would mean that on average, there are 50 people for every 1 TV in the country. a) Create the scatterplot of people per TV(x) and Life Expectancy(y). Describe the scatterplot. Negative Curve Moderately Strong(r = ) No outliers b) Find the LSRL, r, and r 2 values. Add the line to the plot. y = x r = R 2 = 64.6% c) Create the residual plot and sketch it below. d) Is the linear model appropriate for this data? Why? No. There is an obvious curve in the residual plot. e) Is there an association between the two variables? Why? Yes. There is a clear pattern in the scatterplot.

8 f) Is there a cause and effect relationship between the amount of people per TV and the life expectancy? There is no evidence of a cause and effect relationship. g) What is R 2? Interpret this in the context of the problem. R 2 = 64.6%; 64.6% of the variability in Life Expectancy can be explained by variations in People per TV. h) So what percent of life expectancy is due to OTHER variables (things other than people per TV)? 35.4% i) Suggest some confounding variables that are associated with both a country s life expectancy and the prevalence of TVs in the country. (what else could be causing life expectancy to go up and people per TV to go down at the same time??) - Amount of money the people have - Access to medical facilities -