Statistics – Ch 3.2A Notes Name: ___________________________ Least-Squares Regression

Regression Line Everyone knows that cars and trucks lose value the more they are driven. Can we predict the price of a used Ford F-150 SuperCrew 4X4 if we know how many miles it has on the odometer? A random sample of 16 used Ford F-150 SuperCrew 4X4s was selected from among those listed for sale at autotrader.com. The number of miles driven and price (in dollars) were recorded for each of the trucks. Here are the data. Miles Price 70,583 21,994 129,484 9,500 29,932 29,875 29,953 41,995 24,495 41,995 75,678 28,986 8,359 31,891 4,447 37,991 34,077 34,995 58,023 29,988 44,447 22,896 68,474 33,961 144,162 16,883 140,776 20,897 29,397 27,495 131,385 13,997 Linear (straight-line) relationships between two quantitative variables are common and easy to understand. A regression line summarizes the ________________________ between two variables, but only in settings where one of the variables helps explain or predict the other. A ______________________________ is a line that describes how a response variable 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. A regression line is a _________________ for the data, much like density curves. The equation of a regression line gives a compact mathematical description of what this model tells us about the relationship between the response variable y and the explanatory variable x.

Suppose that y is a response variable and x is an explanatory variable.

A regression line relating y to x has an equation of the form

ŷ = a + bx

In this equation,

• ŷ (read “y hat”) is the _______________________of the response variable y for a given value

of the explanatory variable x.

• b is the _________________, the amount by which y is predicted to change when x increases

by one unit.

• a is the __________________, the predicted value of y when x = 0.

Key

relationship

Regression Line

model

predictedvalue

slope

g intercept

Statistics – Ch 3.2A Notes Name: ___________________________ Least-Squares Regression

Example Problem: Interpreting slope and y-intercepts The equation of the regression line is shown 1. Identify the slope and y-intercept of the

regression line.

2. Interpret each value in context.

3. Use the regression line to predict the price for a Ford F-150 with 100,000 miles driven.

_____________________________ is the use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line. Such predictions are often not accurate.

Don’t make predictions using values of x that are much larger or much smaller than those that actually appear in your data.

Practice Problem: 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 linear relationship. The linear regression equation 𝑊𝑒𝑖𝑔ℎ𝑡̂ = 100 + 40(𝑡𝑖𝑚𝑒) models the data fairly well.

1. What is the slope of the regression line? Explain what this means in context.

2. What is the y-intercept? Explain what this means in context

3. Predict the rat’s weight after 16 weeks. Show your work.

price = 38257 0.1629(miles driven)

Slope o b o 1629

Y int o a 38,257

b Theprice of ausedfordf 150 ispredictedtogodownbyo1629dollars 164foreachadditionalmilethetruckhasbeendrivena Thepredictedpriceofaford f Iso thathasbeendrivenzeromiles is 38,257 Newtruck

price 38257 o1629 toooooprice 21967

Extrapolation

b 40j foreachadditionalweek we predictthat a ratwillgain40grams ofweight

a loo Thepredictedweightof a new bornrat is 100grams

weight 100 t 4046weight 740grams

Statistics – Ch 3.2A Notes Name: ___________________________ Least-Squares Regression

4. Should you use this line to predict the rat’s weight at age 2 years? Use the equation to make the prediction and think about the reasonableness of the result. (there are 454 grams in a pound)

Residuals In most cases, no line will pass exactly through all the points in a scatterplot. A good regression line makes the vertical distances of the points from the line as small as possible. A _________________ is the difference between an observed value of the response variable and the value predicted by the regression line.

residual = ___________ – ___________ Other Names for Regression Line:

x Least Squares Regression Line x LSRL x Linear Model x Regression Equation

Practice Problem: Finding a Residual Find and interpret the residual for the Ford F-150 that had 70, 583 miles driven and a price of $21, 994. Least Squares Regression Line Different regression lines produce different residuals. The regression line we want is the one that

minimizes the sum of the squared residuals.

The least-squares regression line of y on x is the line that makes the sum of the ________________

____________________ as small as possible.

driven) miles(1629.038257price �

2years 104weeks This is unreasonable and isweight loot 40404 theresultofextrapolationweight 4,260grams

or94lbs

residualobserved predicted y

X Y a observer

price 382570.162970.583 Theactualprice of this truckprice 26,759 residualsobserved Predicted was 4 65 lowerthanexpectedpredicted residual 21,994 26,759 basedonitsmileageTheactualresidual D 4,765 pricemightbeloweras a result

ofmanyotherfactors

squaredresiduals