THE SAT ESSAY:
IS LONGER BETTER?
In March of 2005, Dr. Perelmen from MIT reported,
“It appeared to me that regardless of what a student
wrote, the longer the essay, the higher the score. If you
just graded them based on length without ever reading
them, you’d be right over 90% or the time.” Analyze the
data and use it to respond to Dr. Perelmen’s claim.
Words
Score
Words
Score
Words
Score
Words
Score
460
6
201
4
403
5
128
2
422
6
168
4
401
6
67
1
402
5
156
3
388
6
697
6
365
5
133
2
320
5
387
6
357
6
114
2
258
4
355
5
278
5
108
1
236
4
337
5
236
4
100
1
189
3
325
4
272
4
150
2
135
3
SAT Essay Scores
9
y = 0.0104x + 1.1728
R² = 0.7753
8
7
6
Score
5
Score
4
Linear (Score)
3
2
1
0
0
100
200
300
400
500
Number of words per essay
600
700
800
SAT ESSAY
7
y = 0.013x + 0.5697
R² = 0.8764
6
5
SCORE
4
Series1
3
Linear (Series1)
2
1
0
0
50
100
150
200
250
300
WORDS PER ESSAY
350
400
450
500
LSRL –Least Squares
Regression Line
O The line that minimizes the distance from
each data point to the linear model.
LSRL –Least Squares
Regression Line
O Model for the data
O Helps us predict y given an x value.
ŷ  a  bx
O 𝑦 represents the predicted response value
Ox
represents the explanatory value
O a represents the predicted y-int.
Ob
represents the slope (y/x)
Does Fidgeting Keep You Slim?
NEA
change
(cal)
-94
-57
-29
135 143
151
245
355
Fat Gain
(kg)
4.2
3.0
3.7
2.7
3.2
3.6
2.4
1.3
NEA
change
(cal)
392
473 486
535 571
580
620 690
Fat Gain
(kg)
3.8
1.7
2.2
0.4
2.3
1.6
1.0
(NEA) Non-Exercise Activity
1.1
O Find the regression line.
NEA vs Fat Gain
4.5
4
3.5
3
Fat gain (kg)
ŷ
2.5
Fat Gain (kg)
2
Linear (Fat Gain (kg))
1.5
1
y = -0.0034x + 3.5051
R² = 0.6061
0.5
0
-200
-100
0
100
200
300
NEA (cal)
400
500
600
700
800
O Interpret each value (y-int &
slope) in context.
O Predict: if NEA increases to 400
calories, what will the fat gain be?
ŷ
O What about if NEA increases
to 1500 cal?
O Interpolation – the use of a regression
line for prediction within the interval of
values of explanatory variable x.
O A good predictor.
O Extrapolation – the use of a regression
line for prediction far outside the interval of
values of explanatory variable x.
O Often not accurate
Example 2
O Some data were collected on the weight of a
male white laboratory rat for the first 25
weeks after its birth. A scatterplot of the
weight (g) and time since birth (weeks)
shows a fairly strong, positive linear
relationship. The linear regression equation
𝑤𝑒𝑖𝑔ℎ𝑡 = 100 + 400(𝑡𝑖𝑚𝑒)
O What is the slope of the regression line in
context?
Example 2
O Some data were collected on the weight of a
male white laboratory rat for the first 25
weeks after its birth. A scatterplot of the
weight (g) and time since birth (weeks)
shows a fairly strong, positive linear
relationship. The linear regression equation
𝑤𝑒𝑖𝑔ℎ𝑡 = 100 + 400(𝑡𝑖𝑚𝑒)
O Predict the rat’s weight after 16 weeks.
Example 2
O Some data were collected on the weight of a
male white laboratory rat for the first 25
weeks after its birth. A scatterplot of the
weight (g) and time since birth (weeks)
shows a fairly strong, positive linear
relationship. The linear regression equation
𝑤𝑒𝑖𝑔ℎ𝑡 = 100 + 400(𝑡𝑖𝑚𝑒)
O What about 2 years?
Residuals
O The difference between an observed value
of response variable and value predicted by
the regression line..
residual  y  yˆ
O
e represents residual
O 𝑦 represents the predicted response value
O y
represents the actual
response value
ŷ
Residuals
o Negative residual
means the model
OVER PREDICTS the
y value.
o Positive residual
means the model
UNDER PREDICTS
the y value.
Example 3
O Find and interpret the residual for a person
who overeats and changes their NEA by 580
calories.
𝑓𝑎𝑡 𝑔𝑎𝑖𝑛 = 3.505 − 0.0034(𝑁𝐸𝐴 𝑐ℎ𝑎𝑛𝑔𝑒)
NEA change
(cal)
-94
-57
-29
135
143
151
245
355
Fat Gain
(kg)
4.2
3.0
3.7
2.7
3.2
3.6
2.4
1.3
NEA change
(cal)
392
473
486
535
571
580
620
690
Fat Gain
(kg)
3.8
1.7
1.6
2.2
1.0
0.4
2.3
1.1
EXIT TICKET
Write down the LSRL for the SAT question.
Describe the slope in context of the data.
Describe the y-intercept in context of the data.
Explain why it doesn’t make sense.
Predict what your score would be if you wrote 300
words. How about 700 words?
SAT Essay Scores
9
y = 0.0104x + 1.1728
R² = 0.7753
8
7
6
Score
5
Score
4
Linear (Score)
3
2
1
0
0
100
200
300
400
500
Number of words per essay
600
700
800