Regression x on y; interpretation practice questions.

A researcher has bivariate data on hours of revision y and quiz score x. State which regression line should be used to estimate a value of x from a known value of y.

• The regression line of y on x
• The regression line of x on y
• Either line, since they give identical predictions
• Neither; regression cannot be used to estimate x

Answer: The regression line of x on y

To predict x from y, the regression line of x on y must be used, because it is fitted by minimizing errors in the x-direction.

For a set of bivariate data, both the regression line of y on x and the regression line of x on y always pass through the same point. State this point.

• (0, 0)
• (median of x, median of y)
• (x̄, ȳ)
• (mode of x, mode of y)

Answer: (x̄, ȳ)

Both least-squares regression lines pass through the mean point (x̄, ȳ) of the data.

The regression line of x on y for a data set is x = 2.5y − 4. Calculate the predicted value of x when y = 6.

• 11
• 4
• 19
• 6.5

Answer: 11

Substitute y = 6 directly: x = 2.5 × 6 − 4 = 15 − 4 = 11.

In which situation would using the regression line of y on x to predict a value of x give the same result as using the line of x on y?

• Whenever the sample is large
• Whenever the gradient is positive
• When the correlation coefficient r = 0
• When r = ±1 (perfect correlation)

Answer: When r = ±1 (perfect correlation)

The two regression lines coincide only when the correlation is perfect (r = ±1); otherwise each line must be used for its own direction of prediction.

The regression line of x on y is written x = a + by. State the correct interpretation of the coefficient b.

• x increases by exactly b units every time y increases by 1
• On average, x changes by b for each 1-unit rise in y
• y changes by b units for each 1-unit increase in x
• b is the value of x when y = 0

Answer: On average, x changes by b for each 1-unit rise in y

The gradient describes the average (expected) change in x per unit increase in y; individual data points do not change by exactly b.

A regression line of x on y was found from data with y values between 10 and 30. Suggest why a prediction of x for y = 55 should be treated as unreliable.

• The sample must have been too small
• The gradient of the line must be negative
• y = 55 is beyond the data range: extrapolation
• The x on y line can never be used to predict x

Answer: y = 55 is beyond the data range: extrapolation

Predictions are only considered reliable within the range of the observed data; using y = 55 extrapolates beyond it, where the linear model may not hold.