1. Ch. 12 More about regression
Ch Transforming to Achieve Linearity
Ch. 12 More about regression

2. L1 → Length
L2 → Weight
LinReg for L1 and L2
𝑤𝑒𝑖𝑔ℎ𝑡 =− (𝐿𝑒𝑛𝑔𝑡ℎ)
𝑤𝑒𝑖𝑔ℎ𝑡 =−
= grams
Look at the data, is this a good prediction? No

3. Resid (Obs − Exp): L4 → L2 − L3
Pred: L3 → − (L1)
Res
Resid (Obs − Exp): L4 → L2 − L3
Resid Plot: stat plot → L1 vs L4
Length
curved
not linear
transform
powers
roots
logarithms
Weight is related to volume.
weight = a ⋅ (length)3
Volume is usually proportional to (length)3.

4. L3 → L1 3 stat plot → L3 vs L2 𝑤𝑒𝑖𝑔ℎ𝑡 =4.07+.0147 𝑙𝑒𝑛𝑔𝑡ℎ 3
Weight
stat plot → L3 vs L2
(Length)3
𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑙𝑒𝑛𝑔𝑡ℎ 3
LinReg for L3 and L2

5. Much better prediction 𝑤𝑒𝑖𝑔ℎ𝑡 =4.07+.0147 24 3 =207.3 grams
𝑤𝑒𝑖𝑔ℎ𝑡 =
=207.3 grams
𝑟=.997
What does this mean?
correlation coefficient
Strong positive relationship for predicted weight and (length)3.
Pred: L4 → (L3)
Res
Resid (Obs − Exp): L5 → L2 − L4
(Length)3
Resid Plot: stat plot → L3 vs L5
Random scatter = good!

6. Now use LinRegTTest to check each value
𝑤𝑒𝑖𝑔ℎ𝑡 = 𝐿𝑒𝑛𝑔𝑡ℎ 3
use (length)3 list
𝑏−𝛽 S E 𝑏
= .0147−
𝑠 𝑠 𝑥 𝑛−1
𝑠 𝑥 =
4.07
.563
always two-sided
.0147
.00024
61.07
18.84
99.52%
𝑡cdf .59, 9999, 18 ×2
𝑟𝑒𝑠𝑖 𝑑 2 𝑛−2
𝑟 2
𝑡cdf(61.07, 9999, 18)×2
Now use LinRegTTest to check each value

7. Same for ln (natural log)
Linear function.
Already in linear form.
Logarithmic function.
Already in linear form.
Exponential function.
Power function.
log 𝑦 = log (𝑎 𝑏 𝑥 )
log 𝑦 = log (𝑎 𝑥 𝑏 )
log 𝑦 = log 𝑎 + log 𝑏 𝑥
log 𝑦 = log 𝑎 + log 𝑥 𝑏
log 𝑦 = log 𝑎 + 𝑥log 𝑏
log 𝑦 = log 𝑎 + 𝑏log 𝑥
linear form
linear form

8. Use your calculator and do LinReg for log 𝑥 and log 𝑦 lists
length
weight
log(L1)
log(L2) 𝑥 𝑦
log⁡(𝑥)
log⁡(𝑦) no no no
yes!
power
𝑦=𝑎 𝑥 𝑏
log 𝑤𝑒𝑖𝑔ℎ𝑡 =− log 𝑙𝑒𝑛𝑔𝑡ℎ
Use your calculator and do LinReg for log 𝑥 and log 𝑦 lists
Yes! Random scatter means there’s a linear relationship.
Pred: L5 → − (L3)
Res
Resid (Obs − Exp): L6 → L4 − L5
log(length)
Resid Plot: stat plot → L3 vs L6

9. log 𝑤𝑒𝑖𝑔ℎ𝑡 =−1.899+3.04 log 𝑙𝑒𝑛𝑔𝑡ℎ
𝑤𝑒𝑖𝑔ℎ𝑡 =0.0126⋅ 𝑙𝑒𝑛𝑔𝑡ℎ 3.04
𝑦=𝑎 𝑥 𝑏
weight
LSRL
On your calculator, put this equation into Y1 and graph it with the original scatterplot (𝑥 vs 𝑦).
length
On the AP Exam, you don’t need to know the algebraic properties of logs. However, you should know how to use calculations to transform data using logs and how to make predictions using the LSRL.

10. 𝑦 =𝑎+𝑏𝑥 𝑦 =2.3+4.1𝑥 𝑥 vs 𝑦 𝑦 =𝑎+𝑏𝑙𝑜𝑔 𝑥 𝑦 =2.3+4.1𝑙𝑜𝑔 𝑥 log(𝑥) vs 𝑦
ln⁡(𝑥)
𝑦 =𝑎+𝑏𝑙𝑜𝑔 𝑥
𝑦 = 𝑙𝑜𝑔 𝑥
log(𝑥) vs 𝑦
ln⁡(𝑦)
𝑦 =𝑎 𝑏 𝑥
𝑦 = 𝑥
𝑥 vs log 𝑦
ln⁡(𝑥)
ln⁡(𝑦)
𝑦 =𝑎 𝑥 𝑏
𝑦 =2.3 𝑥 4.1
log 𝑥 vs log 𝑦

11. However, this is the incorrect line of best fit.
L1 → X
L2 → Y
Nope!
Looks exponential
𝑡𝑟𝑎𝑛𝑠 =− (𝑦𝑒𝑎𝑟𝑠)
𝑟=0.669
However, this is the incorrect line of best fit.
We need to transform the data.

12. Since we plot 𝑥 vs ln⁡(𝑦), we use the form ln 𝑦 =𝑎+𝑏𝑥.
Yes!
ln(transistors)
years since 1970
LinReg for L1 and LNY
Since we plot 𝑥 vs ln⁡(𝑦), we use the form ln 𝑦 =𝑎+𝑏𝑥.
ln⁡( 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 )= (𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 1970)
𝑟=0.994
much better!

13. ln⁡( 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 )=7.065+0.366(𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 1970)
2045−1970=75
ln⁡( 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 )=34.515
𝑒 ln⁡( 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 ) = 𝑒
𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 =9.77× 10 14
977 trillion
What do we need to be careful about?
Extrapolation!

14. Residual Plot (𝒙 vs resid): stat plot → L1 vs L4
𝑡= 𝑏−𝛽 S E 𝑏
Constant
7.065
# yrs since 1970
0.366
34.91
5.14× 10 −15
0.544
98.86%
𝑡cdf(34.91, 9999, 14)×2
𝑟𝑒𝑠𝑖 𝑑 2 𝑛−2
𝑟 2
Predicted values:
L3 → (L1)
Res
Residuals (Obs − Exp):
years since 1970
L4 → LNY − L3
Residual Plot (𝒙 vs resid):
stat plot → L1 vs L4

15. Make a list called LOGY → log(L1)
Yes!
Make a list called LOGY → log(L1)
Plot L1 vs LOGY
log⁡⁡( 𝑡𝑟𝑎𝑛𝑠 )= (𝑦𝑒𝑎𝑟𝑠)
LinReg → L1 and LOGY
log⁡⁡( 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 )= (𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 1970)
log⁡⁡( 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 )= (75)
log⁡⁡( 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 )=14.993
10 log 𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 =
𝑡𝑟𝑎𝑛𝑠𝑖𝑠𝑡𝑜𝑟𝑠 =9.84× 10 14
984 trillion
Very close to the prediction when using natural log