Free WACE Mathematics Applications Flashcard Review

Identify a problem; pose a statistical question; collect or obtain data; analyse the data; interpret and communicate the results. In an exam answer, name the step you are performing and always return to the original question in context.

A table that cross-classifies each individual by two categorical variables, one giving the rows and one giving the columns. Each cell holds the number of individuals in that combination; the row sums, column sums and grand total (the marginal totals) sit in the margins and must add to the total number of individuals.

Percentage in the direction of the explanatory variable, so that the percentages within each category of the explanatory variable add to 100%. If the explanatory variable defines the columns, use column percentages; if it defines the rows, use row percentages. Comparing percentages in the wrong direction can suggest an association that is not there.

Compare the percentage in the same response category across the categories of the explanatory variable and quote the figures. For example: 'There is an association: 68% of smokers reported a persistent cough compared with 21% of non-smokers.' If the percentages are close to equal across the categories, state that there is no evidence of an association.

The variable used to explain or predict the other variable — the one plotted on the horizontal axis of a scatterplot and used as x in the least-squares line. Choosing it is a decision about the context, not something the data itself decides.

The variable being explained or predicted. It is plotted on the vertical axis of a scatterplot and is the y in the fitted least-squares line. Swapping the response and explanatory variables changes the least-squares line, so identify them before you fit anything.

A plot of one point for each individual, with the explanatory variable on the horizontal axis and the response variable on the vertical axis. It is used to see whether an association between two numerical variables is present and to judge its direction, form and strength before any line is fitted.

Positive if the response variable tends to increase as the explanatory variable increases (points trending upwards to the right); negative if the response variable tends to decrease as the explanatory variable increases. If there is no consistent upward or downward trend, report that there is no association.

Whether the pattern in the points is linear (well described by a straight line) or non-linear (curved). Form must be checked first: r and the least-squares line are only meaningful measures when the form is linear.

How closely the points cluster about the underlying pattern — described as strong, moderate or weak. A strong association has points tightly grouped along the trend; a weak one has points widely scattered. Always report direction, form and strength together, in context.

A number between −1 and 1 that measures the strength and direction of a linear association between two numerical variables. r = 1 is perfect positive linear, r = −1 is perfect negative linear, r = 0 is no linear association. The syllabus expects r to be calculated using technology; it has no units and does not change if you swap which variable is x.

It tells you there is no linear association between the two variables. It does not mean the variables are unrelated — a strong curved (non-linear) relationship such as a parabola can still give r close to 0, which is why you must look at the scatterplot as well as the number.

Use the sign for direction and the size for strength, with the commonly used guide: |r| ≥ 0.75 strong, 0.5 ≤ |r| < 0.75 moderate, 0.25 ≤ |r| < 0.5 weak. These bands are a convention rather than a fixed rule, so always pair the label with the actual value, e.g. 'a strong negative linear association (r = −0.87)'.

The square of the correlation coefficient, r², usually expressed as a percentage. It gives the proportion of the variation in the response variable that can be explained by the variation in the explanatory variable. The remaining 1 − r² is variation explained by other factors.

'r² = 0.72, so 72% of the variation in race time can be explained by the variation in hours of training.' Name both variables, use the word 'variation' twice, and say 'explained by' — 'caused by' is wrong, and '72% of race times' is wrong because it is the variation that is explained.

r = −0.8. Take the square root of r² (√0.64 = 0.8) and attach the sign of the slope of the least-squares line, which is negative here. r² alone never tells you the direction — you must read it from the plot or the sign of the gradient.

y = a + bx, with the intercept a written first and the slope b as the coefficient of x. Fit it using technology to the numerical bivariate data, and always rewrite it with the actual variable names, e.g. cost = 25 + 3.4 × distance, before interpreting.

'On average, for each increase of one <unit of x> in <x>, <y> increases (or decreases if b is negative) by <units of y>.' Include the units of both variables and the words 'on average', and never claim the increase in x causes the change in y.

'When <x> is zero, the predicted <y> is <a> <units>.' Add whether this is meaningful: if x = 0 lies well outside the range of the data, or is impossible in context (e.g. a car of zero age), the intercept has no sensible practical interpretation and you should say so.

Residual = actual value − predicted value, for one data point: the vertical distance from the point to the least-squares line. A positive residual means the line under-predicts that point (the point sits above the line); a negative residual means the line over-predicts it.