We are interested in exploring the relationship between the weight of a vehicle and its
fuel efficiency (gasoline mileage). The data in the table show the weights, in pounds, and fuel efficiency, measured in miles per gallon, for a sample of 12 vehicles.
Weight
Fuel Efficiency
2715
24
2570
28
2610
29
2750
38
3000
25
3410
22
3640
20
3700
26
3880
21
3900
18
4060
18
4710
15
Graph a scatterplot of the data.
Find the correlation coefficient and determine if it is significant.
Find the equation of the best fit line.
Write the sentence that interprets the meaning of the slope of the line in the context of the data.
What percent of the variation in fuel efficiency is explained by the variation in the weight of the vehicles, using the regression line? (State your answer in a complete sentence in the context of the data.)
Accurately graph the best fit line on your scatterplot.
For the vehicle that weights 3000 pounds, find the residual (y-yhat). Does the value predicted by the line underestimate or overestimate the observed data value?
Identify any outliers, using either the graphical or numerical procedure demonstrated in the textbook.
The outlier is a hybrid car that runs on gasoline and electric technology, but all other vehicles in the sample have engines that use gasoline only. Explain why it would be appropriate to remove the outlier from the data in this situation.
Remove the outlier from the sample data. Find the new correlation coefficient, coefficient of determination, and best fit line.
Compare the correlation coefficients and coefficients of determination before and after removing the outlier, and explain in complete sentences what these numbers indicate about how the model has changed.
r = -0.8, significant
yhat = 48.4-0.00725x
For every one pound increase in weight, the fuel efficiency tends to decrease (or is predicted to decrease) by 0.00725 miles per gallon. (For every one thousand pounds increase in weight, the fuel efficiency tends to decrease by 7.25 miles per gallon.)
64% of the variation in fuel efficiency is explained by the variation in weight using the regression line.
yhat=48.4-0.00725(3000)=26.65 mpg. y-yhat=25-26.65=-1.65. Because yhat=26.5 is greater than y=25, the line overestimates the observed fuel efficiency.
(2750,38) is the outlier. Be sure you know how to justify it using the requested graphical or numerical methods, not just by guessing.
yhat = 42.4-0.00578x
Without outlier, r=-0.885, rsquare=0.76; with outlier, r=-0.8, rsquare=0.64. The new linear model is a better fit, after the outlier is removed from the data, because the new correlation coefficient is farther from 0 and the new coefficient of determination is larger.
The four data sets below were created by statistician Francis Anscomb. They show why it is important to examine the scatterplots for your data, in addition to finding the correlation coefficient, in order to evaluate the appropriateness of fitting a linear model.
Set 1
Set 2
Set 3
Set 4
x
y
x
y
x
y
x
y
10
8.04
10
9.14
10
7.46
8
6.58
8
6.95
8
8.14
8
6.77
8
5.76
13
7.58
13
8.74
13
12.74
8
7.71
9
8.81
9
8.77
9
7.11
8
8.84
11
8.33
11
9.26
11
7.81
8
8.47
14
9.96
14
8.10
14
8.84
8
7.04
6
7.24
6
6.13
6
6.08
8
5.25
4
4.26
4
3.10
4
5.39
19
12.50
12
10.84
12
9.13
12
8.15
8
5.56
7
4.82
7
7.26
7
6.42
8
7.91
5
5.68
5
4.74
5
5.73
8
6.89
a. For each data set, find the least squares regression line and the correlation coefficient. What did you discover about the lines and values of r?
For each data set, create a scatter plot and graph the least squares regression line. Use the graphs to answer the following questions:
For which data set does it appear that a curve would be a more appropriate model than a line?
Which data set has an
influential point (point close to or on the line that greatly influences the best fit line)?
Which data set has an
outlier (obviously visible on the scatter plot with best fit line graphed)?
Which data set appears to be the most appropriate to model using the least squares regression line?
a. All four data sets have the same correlation coefficient r=0.816 and the same least squares regression line yhat=3+0.5x
According to the data reported by the New York State Department of Health regarding West Nile Virus
(http://www.health.state.ny.us/nysdoh/westnile/update/update.htm) for the years 2000-2008, the least squares line equation for the number of reported dead birds (
) versus the number of human West Nile virus cases (
) is
. If the number of dead birds reported in a year is 732, how many human cases of West Nile virus can be expected?
The next three questions refer to the following data: (showing the number of hurricanes by category to directly strike the mainland U.S. each decade) obtained from
www.nhc.noaa.gov/gifs/table6.gif A major hurricane is one with a strength rating of 3, 4 or 5.
Decade
Total Number of Hurricanes
Number of Major Hurricanes
1941-1950
24
10
1951-1960
17
8
1961-1970
14
6
1971-1980
12
4
1981-1990
15
5
1991-2000
14
5
2001 – 2004
9
3
Using only completed decades (1941 – 2000), calculate the least squares line for the number of major hurricanes expected based upon the total number of hurricanes.
The data for 2001-2004 show 9 hurricanes have hit the mainland United States. The line of best fit predicts 2.83 major hurricanes to hit mainland U.S. Can the least squares line be used to make this prediction?
No, because 9 lies outside the independent variable values
Yes, because, in fact, there have been 3 major hurricanes this decade
No, because 2.83 lies outside the dependent variable values
Yes, because how else could we predict what is going to happen this decade.
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