1. 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.

2. Words
Score
460 6
201 4
403 5
128 2
422
168
401 67 1
402
156 3
388
697
365
133
320
387
357
114
258
355
278
108
236
337
100
189
325
272
150
135

5. LSRL –Least Squares Regression Line
The line that minimizes the distance from each data point to the linear model.

6. LSRL –Least Squares Regression Line
Model for the data
Helps us predict y given an x value.

7. b represents the slope (y/x)
𝑦 represents the predicted response value
x represents the explanatory value
a represents the predicted y-int.
b represents the slope (y/x)

8. 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
1.6
2.2
1.0
0.4
2.3
1.1
(NEA) Non-Exercise Activity

9. Find the regression line.

10. Interpret each value (y-int & slope) in context.
y = x

11. What about if NEA increases to 1500 cal?
Predict: if NEA increases to 400 calories, what will the fat gain be?
y = x
What about if NEA increases to 1500 cal?

12. Interpolation – the use of a regression line for prediction within the interval of values of explanatory variable x.
A good predictor.
Extrapolation – the use of a regression line for prediction far outside the interval of values of explanatory variable x.
Often not accurate

13. Example 2
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 = (𝑡𝑖𝑚𝑒)
What is the slope of the regression line in context?

14. Example 2
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 = (𝑡𝑖𝑚𝑒)
Predict the rat’s weight after 16 weeks.

15. Example 2
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 = (𝑡𝑖𝑚𝑒)
What about 2 years?

16. Residuals
The difference between an observed value of response variable and value predicted by the regression line..

17. e represents residual
𝑦 represents the predicted response value
y represents the actual response value

18. Residuals Negative residual means the model OVER PREDICTS the y value.
Positive residual means the model UNDER PREDICTS the y value.

19. Example 3
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

20. 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?