Introduction to Statistical Thinking

Relates different variables that are measured on the same sample. Regression models are used to describe the effect of one of the variables on the distribution of the other one. The former is called the explanatory variable and the later is called the response.

The effect of a numeric explanatory variable on the distribution of a numeric response is described in terms of a linear trend.

A plot that presents the data in a pair of numeric variables. The axes represents the variables and each point represents an observation.

A coefficient of a linear equation. Equals the value of \(y\) when the line crosses the \(y\)-axis.

A coefficient of a linear equation. The change in the value of \(y\) for each unit change in the value of \(x\text{.}\) A positive slope corresponds to an increasing line and a negative slope corresponds to a decreasing line.

A measures the joint variability of two numeric variables. It is equal to the sum of the product of the deviations from the mean, divided by the number of observations minus 1.

The residual differences between the values of the response for the observation and the estimated expectations of the response under the regression model (the predicted response).

is the difference between 1 and the ratio between the variance of the residuals from the regression and the variance of the response. Its value is between 0 and 1 and it represents the fraction of the variability of the response that is explained by the regression line.

\(y = a + b \cdot x\text{.}\)

\(\frac{\mbox{Sum of products of the deviations}}{\mbox{Number of values in the sample}-1} = \frac{\sum_{i=1}^n (y_i-\bar y)(x_i - \bar x)}{n-1}\text{.}\)

\(b = \mbox{Covariance}(x,y)/\mbox{Var}(x)\text{.}\)

\(a = \bar y - b\bar x\text{.}\)

\(\Expec(Y_i) = a + b \cdot x_i\text{,}\) \(a\) and \(b\) population parameters.

\(y_i - (a + bx_i)\text{,}\) \(a\) and \(b\) estimated from the data.

\(\sum_{i=1}^n(y_i - (a + bx_i))^2/(n-2)\text{,}\) \(a\) and \(b\) estimated from the data.

\(1 - \sum_{i=1}^n(y_i - (a + bx_i))^2/\sum_{i=1}^n(y_i - \bar y)^2\text{,}\) \(a\) and \(b\) estimated from the data.