The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average...
Transcript of The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average...
![Page 1: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/1.jpg)
1
The R.M.S. Error
Regression for Prediction
![Page 2: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/2.jpg)
2
Prediction Errors
y
x
Regression Line
Error or “residual”
![Page 3: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/3.jpg)
3
A Measure of the Size of the Errors: RMS Error
RMS Error =
(error)2 + (error)2 + ... + (error)2
number of errors
.
![Page 4: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/4.jpg)
4
10 15 20 25 30 35 40 45 50 55 600.5
1
1.5
2
2.5
3
3.5
4
4.5
Quantitative GMAT
MB
A G
PA
Quantitative GMAT Predicts MBA GPA? (r = .34)
![Page 5: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/5.jpg)
5
10 15 20 25 30 35 40 45 50 55 600.5
1
1.5
2
2.5
3
3.5
4
4.5
Quantitative GMAT
MB
A G
PA
RMS Error = .46 Regression line plus one RMS error
Regression line minus one RMS error
![Page 6: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/6.jpg)
6
Interpretation of the RMS Error
• It can be shown that the residuals have average = 0. The RMS error is thus their SD.
• Rule of thumb: about 68% of the residuals are smaller in magnitude than one RMS error. About 95% are smaller in magnitude than two RMS errors
![Page 7: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/7.jpg)
7
The RMS error is a measure of the error around the regression line, in the same sense that the SD is a measure of variability around the mean.
![Page 8: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/8.jpg)
8
10 15 20 25 30 35 40 45 50 55 600.5
1
1.5
2
2.5
3
3.5
4
4.5
Quantitative GMAT
MB
A G
PA
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.50
50
100
150
200
250
300
residuals
RMS Error
RMS Error
Scatter Diagram Histogram of Residuals
![Page 9: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/9.jpg)
9
Match the RMS error: .2 1 5
![Page 10: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/10.jpg)
10
Predicting MBA GPA• Using the GMAT, the error measure would be
the RMS error = .46
• Without knowledge of GMAT, the average would be your best prediciton. A measure of the error would be the SD of MBA GPAs. SD = .49
So you don’t gain much by using the GMAT
![Page 11: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/11.jpg)
11
RMS Error, Correlation, the SD of Y: The Picture
Y Y
X X
Low Correlation High Correlation
![Page 12: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/12.jpg)
12
RMS Error, Correlation, and the SD of Y: The Formula
RMS Error = 1 - r2 x SD of Y
![Page 13: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/13.jpg)
13
Residual Plot: Focus on Prediction Errors
0
Residual = Observed minus Predicted
Scatter Diagram Residual Plot
![Page 14: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/14.jpg)
14
Example: Predicting MBA GPA from Quantitative GMAT
10 15 20 25 30 35 40 45 50 55 600.5
1
1.5
2
2.5
3
3.5
4
4.5
Quantitative GMAT
MB
A G
PA
10 15 20 25 30 35 40 45 50 55 60-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Quantitative GMAT
Res
idua
ls
Scatter Diagram Residual Plot
![Page 15: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/15.jpg)
15
Predicting MBA GPA from Undergrad GPA
1 1.5 2 2.5 3 3.5 40.5
1
1.5
2
2.5
3
3.5
4
4.5
Undergrad GPA
MB
A G
PA
1 1.5 2 2.5 3 3.5 4-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Undergrad GPA
Resi
dual
Scatter Diagram Residual Plot
![Page 16: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/16.jpg)
16
Stream Flow Rate versus Depth
Scatter Diagram Residual Plot
![Page 17: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/17.jpg)
17
Inside Vertical Strips
![Page 18: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/18.jpg)
18
SDs in Vertical Strips
10 15 20 25 30 35 40 45 50 55 600.5
1
1.5
2
2.5
3
3.5
4
4.5
Quantitative GMAT
MB
A G
PA
![Page 19: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/19.jpg)
19
Terminology
• Homoscedastic: same RMS errors in each vertical strip.
• Heteroscedastic: different RMS errors in vertical strips
![Page 20: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/20.jpg)
20
A Heteroscedastic Scatter Plot
![Page 21: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/21.jpg)
21
A Football-Shaped Scatter Diagram is Homoscedastic?
![Page 22: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/22.jpg)
22
The Normal Curve Approximation within a Vertical Strip: The Picture
Regression Lin
e
The data in this vertical strip have an average given by the regression line and an SD equal to the RMS error. The normal approximation can be used with this average and SD.
![Page 23: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/23.jpg)
23
The Normal Curve Approximation within a Vertical Strip: The Algorithm
• Find the average in the strip from the regression line
• The SD within the strip is the RMS error
• Convert to standard units using this mean and SD
• Refer to table of normal curve
![Page 24: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/24.jpg)
24
Example
E3: Average height of father = 68 inches; SD = 2.7
Average height of son = 69 inches; SD = 2.7
r = .50
Scatter diagram is football shaped.
Q: What percent of the sons were over 6 feet tall?
6 feet = 72 inches.
Standard Unit =72 - 69
2.7= 1.11
![Page 25: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/25.jpg)
25
From the Table:
-1.10 1.1072.87 %
13.57 %
So about 14 % of the sons are taller than 72 inches
![Page 26: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/26.jpg)
26
E3: Average height of father = 68 inches; SD = 2.7
Average height of son = 69 inches; SD = 2.7
r = .50
Scatter diagram is football shaped.
Q: What percent of the 6 foot fathers had sons over 6 feet tall?
![Page 27: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/27.jpg)
27
1. Find the average height of the sons from the regression line:
A 6 foot father is 72 - 68
2.7= 1.48 SDs
higher than the father average.
![Page 28: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/28.jpg)
28
Son SD
Father SD r x Son SD
Regression Line and SD Line
![Page 29: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/29.jpg)
29
1.48 father SDs
r x 1
.48
son S
Ds
son average + r x 1.48 son SDs
69 + .5 x 1.48 x 2.7 = 71 inches
Point of Averages
![Page 30: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/30.jpg)
30
So the average in this strip is 71 inches. What is the SD?
RMS Error = 1 - r2 x SD of Y
= 1- .52 x 2.7
= 2.33
![Page 31: The R.M.S. Errorrice/oldstat20_S98/Chapt11.pdf · 31 So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33. To answer question, “What percent in](https://reader033.fdocuments.in/reader033/viewer/2022042014/5e73bb531f2d68794d524232/html5/thumbnails/31.jpg)
31
So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33.
To answer question, “What percent in this strip are over 6 feet tall?” use the normal curve.
6 feet = 72 inches
= (72 - 71)/2.33 standard units
= .43 standard units
34.73 32.6 32.6
About 33% of the sons of 6 foot fathers are taller than 6 feet.