Free SACE General Mathematics Flashcard Review

The cycle used to answer a question with data: pose a statistical question, collect or source appropriate data, represent and analyse the data (here, scatter plot, r, r², regression line, residuals), then interpret the results and communicate a conclusion that answers the original question, including the limitations of the model. In bivariate work the analysis stage is where you choose between a linear and an exponential model.

The variable being explained or predicted; it is plotted on the vertical (y) axis and is the y in y = a + bx. Getting this the wrong way round produces a different least squares line and loses the marks for every prediction that follows, so identify it before entering lists into the calculator.

A display of paired bivariate numerical data in which each pair is plotted as a single point (explanatory variable on the horizontal axis, response variable on the vertical axis). For full marks: both axes labelled with the variable name and units, an appropriate uniform scale (which need not start at zero if a break is shown), and every data point plotted accurately.

Direction (positive or negative), form (linear or non-linear) and strength (strong, moderate or weak), each stated in the context of the two named variables. A three-mark 'describe the association' part almost always allocates one mark to each feature, so never write only 'strong positive'.

As the explanatory variable increases the response variable also tends to increase, so the points on the scatter plot slope upwards from left to right and r is positive (b in y = a + bx is also positive). Write it in context, e.g. 'as engine capacity increases, fuel consumption tends to increase'.

Whether the points follow a straight-line pattern (linear form) or a curved pattern (non-linear form). Only a linear form justifies fitting a least squares line; a clearly curved scatter plot, or a curved pattern in the residual plot, points to an exponential model y = a·b^x instead.

By how closely the points cluster about a straight line, judged using Pearson's r and/or the coefficient of determination r² (the scatter plot alone may support the description but the statistic must be quoted). Values of r near ±1 mean strong, near ±0.5 moderate and near 0 weak; quote the actual value to at least three decimal places to justify your wording.

A number between −1 and 1 that measures the direction and strength of a LINEAR association between two numerical variables. Its sign gives the direction and its size gives the strength; r = 1 or −1 means every point lies exactly on a straight line and r = 0 means no linear association. In this course r is found with electronic technology, not by hand.

|r| of about 0.75 to 1 is strong, about 0.5 to 0.75 is moderate, and below about 0.5 is weak; |r| very close to 0 means effectively no linear association. Always quote the actual value with the description, e.g. 'r = 0.912, so there is a strong positive linear association between height and arm span'.

r only measures how close the points are to a STRAIGHT line. Data following a smooth curve, or data containing one extreme outlier, can produce a large |r| even though a linear model is inappropriate. The scatter plot (and the residual plot) is what tells you whether the linear form assumed by r is reasonable.

No — r is the same either way, because it measures the strength and direction of the association, not a prediction rule. The least squares regression equation DOES change when you swap the variables, so the choice of explanatory variable matters for y = a + bx but not for r or r².

At least three decimal places (e.g. r = −0.874, r² = 0.764), unless the question specifies otherwise. Rounding to one or two places can lose an accuracy mark, and any rounding should be done only at the final step — carry the unrounded calculator value through intermediate working.

'r² = 0.836, so 83.6% of the variation in test score can be explained by the variation in the number of hours studied.' Name both variables, state which one has its variation explained (the response variable), and give the percentage — a bare '83.6% of the data fits the line' scores nothing.

A data point that lies well away from the general pattern formed by the rest of the points on the scatter plot. In this course outliers are identified visually from the scatter plot rather than by a numerical rule, and they may be unusual in x, in y, or in the combination of the two.

Only when there is a reasonable justification — for example the value is a recording or measurement error, or the case does not belong to the population being investigated (a commercial vehicle in a survey of family cars). You must state that justification; deleting a point simply because it improves r is not acceptable and should be commented on if a question implies it.

If the outlier lies well off the linear trend, removing it usually moves |r| closer to 1 and increases r², making the association appear stronger. Occasionally an outlier lies far out along the trend line and props r up, so removing it decreases |r|. In the exam always recalculate both values with technology and compare the before and after figures explicitly.

Say what the outlier does to the statistic and to the model, with numbers: e.g. 'With the outlier r = 0.612 (moderate); without it r = 0.941 (strong), and the slope changes from 2.3 to 3.1, so the single point substantially weakens the apparent association and flattens the fitted line.' One mark for the numerical comparison, one for the interpretation in context.

(1) Coincidence — with enough variables, two unrelated quantities can happen to trend together over the period sampled. (2) A third (confounding) variable that drives both, e.g. ice-cream sales and drowning deaths both rise with hot weather. State which explanation you mean and name the plausible third variable in the given context.

A variable not included in the study that influences both the explanatory and the response variable, creating an association between them that is not causal. In an answer you must name a plausible one for the actual context — e.g. 'towns with more parks also have higher average income, and income affects both park provision and residents' health'.

Say that r = 0.93 shows only a strong positive linear association between app use and sleep quality, not that one causes the other: the relationship could be coincidental, could run the other way (people who sleep well keep using the app), or could be explained by a third variable such as an overall change in bedtime routine. A controlled study, not correlation, would be needed to establish causation.