least-squares regression: linear regression section 3.2 reference text: the practice of statistics,...
TRANSCRIPT
![Page 1: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/1.jpg)
Least-Squares Regression:Linear Regression
Section 3.2
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
![Page 2: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/2.jpg)
Warm up/ quiz
• Draw a quick sketch of three scatterplots:– Draw a plot with r ≈ .9– Draw a plot with r ≈ -.5– Draw a plot with r ≈ 0
![Page 3: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/3.jpg)
Today’s Objective
• Regression line, introducing “y-hat”– Predicted value – Slope– Y-intercept
• Extrapolation• Residuals • Least-squares Regression Line
– How to use your calculator effectively for time
y a bx
![Page 4: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/4.jpg)
Your Poster!
• Take a look at your poster: Do you think you could draw a straight line that would go straight through the middle where you have ½ your points above and ½ your points below?– Calculate your line:
• m = y2-y1 / x2-x1 • Point slope form: y – y1 = m ( x - x1)
• In math-land this is known as a “line of best fit”
![Page 5: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/5.jpg)
Regression Line
• In statistics, this is called a regression line!
• A line that describes how a response variables y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x.
![Page 6: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/6.jpg)
Formulas for Regression Line• The RL is linear, so it follows the form y = mx + b
– In Statistics, we say
– In this context, is called the predicted value– WARNING: we are entering “predicting” statistics,
using the “ ” symbol is very important – [story about AP training]
– So is the predicted value– And ‘a’ is the y-intercept, the predicted value of y
when x=0– And ‘b’ is the slope
y
y a bx
y
y
y a bx
![Page 7: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/7.jpg)
Equation of Regression
![Page 8: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/8.jpg)
The Meaning of Slope• In a simple algebraic function like y = 2x + 17,
what is the real meaning of the slope?– For every increase in x of 1 unit, y increases by 2
• In the function y = 2x + 17 what is the meaning of the y intercept?– It is the value y takes on when x = 0
• In statistics if the regression line is = 3.505 - .00344x– What is the slope?– What is the y-intercept?
y
![Page 9: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/9.jpg)
Context
![Page 10: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/10.jpg)
ExtrapolationTake a look at your poster!
• Take a look at the range of your data.
• Your line is linear- so it does on and on even past your data points
• Predict an output value when you input a large number outside your data range
• Put it into context: examples?
• This is whats known as extrapolating!
![Page 11: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/11.jpg)
Extrapolation
• Is the use of a regression line for prediction outside the interval of values of the explanatory variable x used to obtain the line. Such predictions are often not accurate.
• “Just because your line behaves the way it does within the confines, does not mean its gets all squirrely later on! We cant predict the behavior of data to extremes.”
![Page 12: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/12.jpg)
Example
• 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 (in grams) and time since birth (in weeks) shows a fairly strong, positive relationship. The linear regression equation weight = 100 + 40(time) models the data fairly well.
• 1) What is the slope of the regression line? Explain what it means in context
• 2) what’s the y intercept? Explain in context• 3) predict the rat’s weight after 16 weeks, show your
work• 4) Should you use the line to predict the rat’s weight at
age 2 years?
![Page 13: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/13.jpg)
Residuals
• Look at your graph, how far away are your points from your graph?
• Residuals is the difference between an observed value of the response variable and the value predicted by the regression line.
• Residual = observed y – predicted y
![Page 14: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/14.jpg)
Finding a residual
• Find and interpret the residual for the hiker who weighed 187 pounds.
• Regression line: – Pack weight = 16.3 + .0980( 187) = 33.28 lbs– His actual pack weight was 30 pounds.– Residual = observed – predicted– Residual = 30 – 33.28 = -3.28– The (-) sign tells us that the observed is below the
predicted by 3.28 pounds.– Negative: Below predicted, Positive: above Predicted
![Page 15: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/15.jpg)
Least-Squares Regression Line
• We’ve been using the least-squares regression line this whole time!
• We will talk about where the LSRL comes from next class!
• But for now…lets learn how to use our tool to make our Stats crunching FAST!
y a bx
![Page 16: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/16.jpg)
LSRL TI-83/ TI-89
• TI-83– Put your data in L1, and L2– STAT> CALC>#8 >Enter
• Did you know your TI-83 will default to using L1 and L2 as our lists, so as long as you put your data in L1 and L2, you don’t have to tell it!
• TI-89– Statistics/List Editor> F4 (CALC)>#3> #1– Practice next slide!
![Page 17: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/17.jpg)
Practice with your TI Calculator
• You Should get: = 16.3 + .0908x
Body Weight (lbs)
120 187 109 103 131 165 158 116
Backpack Weight (lbs)
26 30 26 24 29 35 31 28
y
![Page 18: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/18.jpg)
Today’s Objective
• Regression line, introducing “y-hat”– Predicted value – Slope– Y-intercept
• Extrapolation• Residuals • Least-squares Regression Line
– How to use your calculator effectively for time
y a bx
![Page 19: Least-Squares Regression: Linear Regression Section 3.2 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore](https://reader033.vdocuments.us/reader033/viewer/2022051415/56649d6e5503460f94a4e86a/html5/thumbnails/19.jpg)
Homework
Worksheet