2 Simple linear regression
A simple linear regression model is a model with a single regressor \(X\) that has a linear relationship with a response \(Y\). The simple linear regression model is \begin {equation} Y = \beta _0 + \beta _1X + \varepsilon \end {equation} where
- \(Y\) is the response variable;
- \(X\) is the regressor variable;
- \(\beta _0\) is the intercept;
- \(\beta _1\) is the slope;
- \(\varepsilon \) is the random error component.
Note. \(Y\) is a random variable, because \(\varepsilon \) is.
Remark. For a given \(X\), the corresponding observation \(Y\) consists of the value \(\, \beta _0 + \beta _1X\,\) plus an amount \(\varepsilon \).
2.1 The model and its assumptions
2.2 Least-squares estimation of the parameters
2.3 Properties of the fitted line
2.4 Properties of the least-squares estimators
2.5 Estimation of \(\sigma ^2\)
2.6 Maximum likelihood estimation
2.7 Sampling distributions under normality
2.8 Hypothesis tests on the slope and intercept
2.9 Analysis of variance
2.10 The coefficient of determination
2.11 Interval estimation
2.12 Correlation
2.13 Practice problems
2.2 Least-squares estimation of the parameters
2.3 Properties of the fitted line
2.4 Properties of the least-squares estimators
2.5 Estimation of \(\sigma ^2\)
2.6 Maximum likelihood estimation
2.7 Sampling distributions under normality
2.8 Hypothesis tests on the slope and intercept
2.9 Analysis of variance
2.10 The coefficient of determination
2.11 Interval estimation
2.12 Correlation
2.13 Practice problems
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