Stat 220 - Summer 2014 - Homework 6
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Plotting Lines
For each line, find the slope and the intercept. Note: the axes do not cross at 0 in each case
2
Plotting Lines
1. Draw lines through the point (2,1) with the slopes: +1, -1, 0
2. Assume you have a line that passes through the point (2,1) and has a slope of 3/1.5. If you start
at (2,1) and move over 2 and up 1, will you be on the line, above the line or below the line?
3. The same, but move over 4 and up 2.
4. The same but move over 6 and up 5.
5. Draw the line with intercept 2 and slope -1.
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Prediction - # 15.1 Fossil bones
Use the equation of the least squares line: humerus length = -3.66 + (1.197× femur length) to
predict the humerus length for a fossil with a femur of 70cm length.
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Prediction - # 15.5 State SAT scores
Figure 14.9 (page 301) plots the average SAT Mathematics score of each states high school seniors
against the proportion of each state’s seniors who took the exam. In addition to two clusters, the plot
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shows an overall roughly straight-line pattern. The least-squares regression line for predicting SAT
Math score from proportion taking is SAT score = 577.6 - (106.9 × proportion taking)
(a) What does the slope b = -106.9 tell us about the relationship between these
variables?
(b) In New York State, the proportion of high school seniors who took the SAT was 0.85
in 2010. Predict their average score. (The actual average score in New York was 505.)
(c) Calculate the residual for 2010.
5
SD Line
For the scatter diagram shown below, say whether it is the solid line or the dashed line which is the
SD line. Argue why.
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SD line
One study on male college students found their average height to be 69 inches, with an SD of 3 inches.
Their average weight was 140 pounds, with an SD of 20 pounds. And the correlation was 0.60. If one
of these people is 72 inches tall, how heavy would he have to be to fall on the SD line?
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SD Line
Using the same data as in exercise 3, say whether each of the following students was on the SD line:
(a) height 75 inches, weight 180 pounds
(b) height 66 inches, weight 130 pounds
(c) height 66 inches. weight 120 pounds
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8
SD and Least Squares Lines
Below are 4 scatterplots, each with a solid line and a dashed line. For each diagram, say which is
the SD line and which is the least squares regression line for y on x.
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Least Squares Regression
A university has made a statistical analysis of the relationship between Math SAT scores (old scores
ranging from 200 to 800) and first-year GPAs (ranging from 0.0 to 4.0), for students who complete the
first year. The results:
Average SAT score=550
Average first-year GPA=2.6
SD=80
SD=0.6
r = 0.4
We want to use the SAT scores an explanatory variable for the first-year GPA. Assume that the
scatter diagram is football shaped. A student is chosen at random, and has an SAT score of 650.
(a) Find the SD line formula
(b) Find the least-squares regression line formula.
(c) Plot both lines on one graph.
(d) Predict this individual’s first-year GPA using the least-squares regression line.
(d) Assume that the student has a first-year GPA of 3.1. Calculate the residual (from
the least-squares regression) for this individual and draw the residual on the plot.
(e) Explain how we can interpret the least-squares regression line slope that you found
in part (b).
(f ) Among students who finish the first year, how well do SAT scores predict first-year
GPA? (Use r2 in your answer.)
(g) The GPA prediction for a student who had an SAT score of 800 is 3.35. Would you
believe this prediction? Why or why not?
(h) Carefully explain why we prefer and commonly use the least-squares regression line
for prediction instead of the SD line.
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