Least-Squares Regression: Inference, Hypothesis Testing, and Confidence Intervals

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Least-Squares Regression Model

Introduction to Least-Squares Regression

The least-squares regression model is a fundamental statistical tool used to describe the linear relationship between an explanatory variable and a response variable. In this context, the Zestimate (predicted selling price) is treated as the explanatory variable, and the Sale Price is the response variable.

Key Terms and Definitions

Explanatory Variable (Predictor): The variable used to explain or predict changes in another variable.
Response Variable: The variable whose changes are being explained or predicted.
Least-Squares Regression Line: The line that minimizes the sum of the squared differences between observed and predicted values.

Example: Zillow Data

The following table presents a sample of three-bedroom homes sold in Seattle, WA. Zestimate is the predicted selling price, and Sale Price is the actual selling price.

Zestimate	Sale Price	Zestimate	Sale Price
371,485	355,000	361,111	350,000
459,767	455,000	300,753	292,000
359,785	420,000	343,141	330,000
494,514	533,210	548,561	594,500
563,303	620,000	426,609	425,000

Additional info: The table is used to illustrate regression analysis and hypothesis testing.

Testing the Significance of the Slope of the Least-Squares Regression Model

Objectives

Use randomization to test the significance of the slope.
State the requirements for inference on the least-squares regression model.
Compute the standard error of the estimate.
Verify that residuals are normally distributed.
Conduct inference on the slope using Student's t-distribution.
Construct a confidence interval for the slope.

Least-Squares Regression Equation

The least-squares regression equation is:

: Response variable
: Explanatory variable
: Intercept
: Slope

For the Zillow data, the estimated intercept and slope are:

Interpreting the Slope and Intercept

Slope (): Represents the change in the response variable for a one-unit change in the explanatory variable.
Intercept (): Represents the predicted value of the response variable when the explanatory variable is zero. In many real-world contexts, interpreting the intercept may not make sense if a zero value for the explanatory variable is not meaningful.

Hypothesis Testing for the Slope

To determine if there is a significant linear relationship between the explanatory and response variables, we test:

Null Hypothesis (): (no linear association)
Alternative Hypothesis (): (linear association exists)

Types of Tests

Test Type	Null Hypothesis	Alternative Hypothesis
Two-Tailed
Left-Tailed
Right-Tailed

Randomization Test

Randomly assign sale prices to each Zestimate.
Draw a scatter diagram and compute the regression line for the randomized data.
Compare the observed slope to the distribution of slopes from random assignments to assess significance.

Requirements for Inference

Requirement 1: For each value of the explanatory variable, the mean of the response variable in the population depends linearly on .
Requirement 2: The response variable is normally distributed with mean and constant standard deviation .

Standard Error of the Estimate

The standard error of the estimate, , quantifies the typical distance that the observed values fall from the regression line. It is calculated as:

where are observed values, are predicted values, and is the sample size.

Verifying Normality of Residuals

Residuals () should be approximately normally distributed.
Check normality using a histogram or normal probability plot.

Conducting Inference on the Slope

To test the significance of the slope, use the following test statistic:

: Sample estimate of the slope
: Hypothesized value (usually 0)
: Standard error of the slope

Steps for Hypothesis Testing

Verify that the explanatory and response variables are quantitative.
State the null and alternative hypotheses.
Build a null model by randomly assigning the response variable.
Estimate the p-value.
State the conclusion.

Constructing a Confidence Interval for the Slope

A confidence interval for the slope of the regression line is given by:

Lower bound: Upper bound:

: Critical value from the t-distribution with degrees of freedom
: Standard error of the slope

Example Application

Use the regression equation to predict the sale price for a given Zestimate.
Assess whether the model can be generalized to other cities (e.g., Chicago) by considering differences in housing markets.

Summary Table: Steps in Regression Inference

Step	Description
1	Verify quantitative variables
2	State hypotheses
3	Build null model (randomization)
4	Estimate p-value
5	State conclusion

Conclusion

Least-squares regression is a powerful method for modeling linear relationships between variables. Proper inference requires checking model assumptions, computing standard errors, and using hypothesis tests and confidence intervals to assess the significance and reliability of the regression slope.

Additional info: These notes expand on the original content by providing definitions, formulas, and structured steps for regression analysis and inference.