Correlation; Pearson's r; scatter diagrams; regression y on x practice questions.

Which measure describes the strength and direction of a LINEAR relationship between two variables?

• the mean
• Pearson's r
• the interquartile range
• the χ² statistic

Answer: Pearson's r

Pearson's r ranges from −1 to +1 and measures linear correlation.

A data set gives Σx = 40, Σy = 60, and mean point (x̄, ȳ). If there are 8 data pairs, state the coordinates of the mean point that the regression line of y on x must pass through.

• (5, 7.5)
• (40, 60)
• (8, 8)
• (7.5, 5)

Answer: (5, 7.5)

x̄ = Σx/n = 40/8 = 5 and ȳ = Σy/n = 60/8 = 7.5, so the mean point is (5, 7.5), through which the least-squares line always passes.

The regression line of y on x for a data set is y = -1.8x + 25. Calculate the predicted value of y when x = 10.

• 43
• 7
• -18
• 23.2

Answer: 7

Substitute x = 10: y = -1.8(10) + 25 = -18 + 25 = 7.

For a set of bivariate data, Pearson's correlation coefficient is r = 0.6. Identify the value of the coefficient of determination r² and its correct interpretation.

• 0.36; proportion of y's variation explained
• 0.6; the strength of the linear relationship
• 0.775; the gradient of the regression line
• 0.36; the probability the correlation is causal

Answer: 0.36; proportion of y's variation explained

r² = 0.6² = 0.36, meaning 36% of the variation in y is explained by the linear regression on x.

A scatter diagram shows points that lie almost exactly on a straight line sloping downwards from left to right. Determine which value of r best describes this data.

• 0.05
• 0.55
• -0.98
• -0.40

Answer: -0.98

A tight downward-sloping pattern indicates a strong negative linear correlation, so r is close to -1; r = -0.98 fits best.

The regression line of y on x is y = 3.2x - 5. It is used to estimate y for x = 200, which is far outside the range of the collected data. State the most appropriate description of this estimate.

• Interpolation, so it is reliable
• Extrapolation, so it may be unreliable
• A residual, so it equals zero
• Correlation, so it proves causation

Answer: Extrapolation, so it may be unreliable

Predicting outside the range of the observed data is extrapolation; the linear trend may not continue, so the estimate can be unreliable.