Quick questions on Bivariate Data and Regression - TCE Mathematics Applications (Tasmania)

When you read a scatterplot, comment on four features: form (linear or non-linear), direction (positive or negative), strength (how tightly the points cluster about a line), and any outliers. A positive direction means $y$ tends to rise as $x$ rises.

Pearson's correlation coefficient $r$ measures the strength and direction of a linear relationship. It always lies between $-1$ and $+1$. Values near $\pm 1$ mean a strong linear pattern; values near $0$ mean little or no linear relationship.

The coefficient of determination is simply $r^2$. It gives the proportion of the variation in the response variable that is explained by the linear relationship with the explanatory variable. If $r = 0.8$ then $r^2 = 0.64$, so about $64\%$ of the variation in $y$ is explained by $x$ (and $36\%$ is due to other factors).

The least-squares line is the straight line $\hat{y} = a + bx$ that minimises the sum of the squared vertical distances from the points to the line. In a TCE exam you usually read $a$ (intercept) and $b$ (slope/gradient) from technology, then interpret them.

When the question gives the summary statistics rather than the raw data, use the least-squares formulae directly. The slope is $$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2},$$and the intercept is$a = \dfrac{\sum y - b\sum x}{n} = \bar{y} - b\bar{x}$. The arithmetic must be careful: compute the numerator and denominator of$b$first, then use that value of$b$ in the intercept formula. The information sheet supplied in the TCE exam carries these formulae, so the marks are for substituting correctly, not for memorising them.