# Simple Linear Regression

**Topics:**Regression analysis, Statistical inference, Linear regression

**Pages:**18 (3021 words)

**Published:**April 21, 2013

Review – Stat 226

Spring 2013

Stat 326 (Spring 2013)

Introduction to Business Statistics II

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Stat 326 (Spring 2013)

Introduction to Business Statistics II

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Review: Inference for Regression

Example: Real Estate, Tampa Palms, Florida Goal: Predict sale price of residential property based on the appraised value of the property Data: sale price and total appraised value of 92 residential properties in Tampa Palms, Florida 1000 900 Sale Price (in Thousands of Dollars) 800 700 600 500 400 300 200 100 0 0 100 200 300 400 500 600 700 800 900 1000 Appraised Value (in Thousands of Dollars)

Review: Inference for Regression

We can describe the relationship between x and y using a simple linear regression model of the form µy = β 0 + β1 x 1000 900 Sale Price (in Thousands of Dollars) 800 700 600 500 400 300 200 100 0 0 100 200 300 400 500 600 700 800 900 1000 Appraised Value (in Thousands of Dollars)

response variable y : sale price explanatory variable x: appraised value relationship between x and y : linear strong positive

We can estimate the simple linear regression model using Least Squares (LS) yielding the following LS regression line: y = 20.94 + 1.069x

Stat 326 (Spring 2013)

Introduction to Business Statistics II

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Stat 326 (Spring 2013)

Introduction to Business Statistics II

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Review: Inference for Regression

Interpretation of estimated intercept b0 : corresponds to the predicted value of y , i.e. y , when x = 0

Review: Inference for Regression

Interpretation of estimated slope b1 : corresponds to the change in y for a unit increase in x: when x increases by 1 unit y will increase by the value of b1

interpretation of b0 is not always meaningful (when x cannot take values close to or equal to zero) here b0 = 20.94: when a property is appraised at zero value the predicted sales price is $20,940 — meaningful?! Stat 326 (Spring 2013) Introduction to Business Statistics II 5 / 47

b1 < 0: y decreases as x increases (negative association) b1 > 0: y increases as x increases (positive association) here b1 = 1.069: when the appraised value of a property increases by 1 unit, i.e. by $1,000, the predicted sale price will increase by $1,069. Stat 326 (Spring 2013) Introduction to Business Statistics II 6 / 47

Review: Inference for Regression

Measuring strength and adequacy of a linear relationship correlation coeﬃcient r : measure of strength of linear relationship −1 ≤ r ≤ 1 here: r = 0.9723

Review: Inference for Regression

Population regression line Recall from Stat 226 Population regression line The regression model that we assume to hold true for the entire population is the so-called population regression line where µy = β0 + β1 x,

coeﬃcient of determination r 2 :

amount of variation in y explained by the ﬁtted linear model 0 ≤ r2 ≤ 1 here: r 2 = (0.9723)2 = 0.9453 ⇒ 94.53% of the variation in the sale price can be explained through the linear relationship between the appraised value (x) and the sale price (y ) Stat 326 (Spring 2013) Introduction to Business Statistics II 7 / 47

µy — average (mean) value of y in population for ﬁxed value of x β0 — population intercept β1 — population slope The population regression line could only be obtained if we had information on all individuals in the population. Stat 326 (Spring 2013) Introduction to Business Statistics II 8 / 47

Review: Inference for Regression

Based on the population regression line we can fully describe relationship between x and y up to a random error term ε y = β0 + β1 x + ε, where ε ∼ N (0, σ)

Review: Inference for Regression

In summary, these are important notations used for SLR: Description x y

Parameters β0 β1 µy ε

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Description

Estimates b0 b1 y e

Description

Introduction to Business Statistics II

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