11 Simple Linear Regression Free Online FlashCards

What does simple linear regression describe?

Simple linear regression describes and predicts the relationship between one quantitative predictor, $X$, and one quantitative response, $Y$.

What is the population simple linear regression model?

The population model is $Y_i=\beta_0+\beta_1x_i+\varepsilon_i$, and its mean response is $E(Y\mid X=x)=\beta_0+\beta_1x$.

What does the least-squares method minimize?

Least squares chooses the line that minimizes $SSE=\sum_{i=1}^{n}(y_i-\hat y_i)^2$, the sum of squared vertical residuals.

What is the least-squares slope formula?

The estimated slope is $b_1=\frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sum (x_i-\bar x)^2}$.

Which point always lies on the fitted regression line?

The fitted line always passes through $(\bar x,\bar y)$, the point formed by the sample means of the predictor and response.

How is a regression slope interpreted?

The slope $b_1$ is the estimated change in predicted $Y$ for a one-unit increase in $X$, on average.

What is a residual, and what does a positive value mean?

A residual is observed minus fitted: $e_i=y_i-\hat y_i$. A positive residual means the observation is above the fitted line.

What does the coefficient of determination $R^2$ measure?

$R^2$ is the proportion of sample variation in the response explained by the fitted regression model: $R^2=1-\frac{SSE}{SST}$.

How are $R^2$ and correlation related?

In simple linear regression with an intercept, $R^2=r^2$. The sign of the correlation agrees with the slope, while $R^2$ is never negative.

How is a zero population slope tested?

The usual two-sided test is $H_0:\beta_1=0$ versus $H_a:\beta_1\ne0$, using $t=\frac{b_1}{SE(b_1)}$ with $n-2$ degrees of freedom.

What is the confidence interval formula for the slope?

A $(1-\alpha)\times100\%$ confidence interval for $\beta_1$ is $b_1\pm t^*SE(b_1)$, using $n-2$ degrees of freedom.

How do mean-response and individual prediction intervals differ?

A confidence interval estimates the mean response at $x^*$; a prediction interval estimates one new individual response and is wider because it includes individual random variation.