Fundamentals of Regression and Hypothesis Testing

📋 Course Outline

1. Regression assumptions and model fit
2. Estimating slope and intercept
3. Slope significance and prediction intervals
4. Correlation tests and rank measures
5. Chi-square and variance tests
6. Choosing the right hypothesis test
7. Sampling error and standard errors

📖 1. Regression assumptions and model fit

🔑 Key Concepts & Definitions

• Normality of residuals: Normality of residuals means the regression errors should follow a normal distribution for inference validity.
• Homoskedasticity assumption: Homoskedasticity means the variance of the residuals stays constant across values of the independent variable.
• Random residual pattern: A random residual pattern means residuals show no systematic structure when plotted against the independent variable.

📝 Essential Points

• Normality in simple linear regression applies to the residuals, not to the dependent and independent variables themselves.
• For large samples, the normality requirement for residuals can be relaxed by the central limit theorem.
• A non-random residual pattern versus the independent variable indicates a violation like heteroskedasticity or non-independence rather than satisfied homoskedasticity.
• Model fit in simple regression uses $R^2=\text{SSR}/\text{SST}$ and $F=\text{MSR}/\text{MSE}$ with $\text{MSR}=\text{SSR}/1$ and $\text{MSE}=\text{SSE}/(n-2)$.
• With $\text{SSR}=90$, $\text{SSE}=110$, $\text{SST}=200$, and $n=22$, $R^2=0.45$ and $F=16.36$.