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QCAA General Maths EA study materials
QCAA General Maths EA study resources

QCAA General Maths EA practice questions

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Question 1Unit 3

A bivariate data set has a correlation coefficient of r = -0.92. Which description best matches the association between the two variables?

  1. Weak negative linear association
  2. Strong positive linear association
  3. Strong negative linear association Correct answer.
  4. No linear association

Explanation

The correlation coefficient r ranges from -1 to 1, and the sign indicates the direction of the linear association. A value of -0.92 is close to -1, which means the relationship is strong and negative, since one variable tends to decrease as the other increases.

Question 2Unit 3

The equation of a least-squares regression line is y = 12.4 - 3.1x. What does the slope of -3.1 mean in context?

  1. The correlation between x and y is 3.1
  2. When x equals zero, y is predicted to equal 3.1
  3. For each one-unit increase in x, y is predicted to decrease by 3.1 Correct answer.
  4. For each one-unit increase in x, y is predicted to increase by 3.1

Explanation

In a regression line of the form y = a + bx, the coefficient b is the slope and gives the predicted change in y for a one-unit increase in x. Here b = -3.1, so a one-unit rise in x predicts a fall of 3.1 in y.

Question 3Unit 3

A least-squares regression line is fitted to a data set. Which statement about the line is true?

  1. It passes through the point whose coordinates are the means of x and y Correct answer.
  2. It must pass through at least two of the data points
  3. It has a slope equal to the correlation coefficient
  4. It always passes through the origin

Explanation

A key property of the least-squares line is that it always passes through the mean point (x̄, ȳ). This follows from the way the intercept is calculated once the slope is found.

Question 4Unit 3

A residual for a data point is calculated as 4.2. What does this indicate?

  1. The observed value is 4.2 above the value predicted by the line Correct answer.
  2. The observed value is 4.2 below the value predicted by the line
  3. The data point lies exactly on the regression line
  4. The correlation coefficient is 4.2

Explanation

A residual is the difference between the observed value and the predicted value, observed minus predicted. A positive residual of 4.2 therefore means the actual value is 4.2 greater than the prediction.

Question 5Unit 3

A time series shows a regular pattern that repeats every 12 months. Which component of the time series does this describe?

  1. Irregular variation
  2. Seasonality Correct answer.
  3. Cyclical variation
  4. Long-term trend

Explanation

Seasonality is a periodic pattern that repeats over a fixed period, such as a year, and is often tied to the calendar. A pattern recurring every 12 months is the classic example of a seasonal component.

Question 6Unit 3

In a least-squares regression, the coefficient of determination is r² = 0.64. What percentage of the variation in the response variable is explained by the regression line?

  1. 8%
  2. 36%
  3. 80%
  4. 64% Correct answer.

Explanation

The coefficient of determination r² gives the proportion of the variation in the response variable that is accounted for by the linear relationship with the explanatory variable. Since r² = 0.64, this is 64 percent.

Question 7Unit 3

A least-squares line is y = 5 + 2x. What is the predicted value of y when x = 10?

  1. 52
  2. 20
  3. 15
  4. 25 Correct answer.

Explanation

Substitute x = 10 into the equation: y = 5 + 2(10) = 5 + 20 = 25. The intercept is added to the product of the slope and x.

Question 8Unit 3

The correlation between two variables is r = 0.85. What is the value of the coefficient of determination?

  1. 0.15
  2. 0.7225 Correct answer.
  3. 0.85
  4. 0.425

Explanation

The coefficient of determination is the square of the correlation coefficient, r². Squaring 0.85 gives 0.7225, which is the proportion of variation explained by the linear model.

Question 9Unit 3

An explanatory variable x has mean 10 and a response variable y has mean 24. A least-squares line is fitted to the data. Which point must lie on the line?

  1. (10, 24) Correct answer.
  2. (0, 0)
  3. (10, 10)
  4. (24, 10)

Explanation

The least-squares regression line always passes through the mean point (x̄, ȳ). With x̄ = 10 and ȳ = 24, the point (10, 24) lies on the line.

Question 10Unit 3

A student computes residuals for a regression and finds the sum of the residuals is zero. Which statement explains this result?

  1. This is expected, because the least-squares line passes through the mean point Correct answer.
  2. This shows every residual is zero
  3. This shows the correlation is exactly zero
  4. This indicates the line was fitted incorrectly

Explanation

For a least-squares line with an intercept, the residuals always sum to zero, which is consistent with the line passing through the mean point. It is a property of the fitted line, not an error.

Question 11Unit 3

A scatterplot of two variables shows a strong curved relationship, but the correlation coefficient is close to zero. Which conclusion is most appropriate?

  1. There is no association between the variables
  2. The correlation was calculated incorrectly
  3. The variables are independent
  4. There is an association, but it is not linear Correct answer.

Explanation

The correlation coefficient measures only the strength of a linear association. A strong curved pattern can produce r near zero even though the variables are clearly related in a nonlinear way.

Question 12Unit 3

A least-squares line is used to predict y from x for values of x well outside the range of the observed data. Which concern applies?

  1. The residuals will all become zero
  2. The slope will change sign
  3. Extrapolation may give unreliable predictions Correct answer.
  4. The correlation will increase

Explanation

Predicting outside the observed range of x is called extrapolation. Because the linear pattern may not continue beyond the data, such predictions are unreliable.

Question 13Unit 3

In a set of bivariate data, the correlation coefficient is r = 0.9. If the units of the response variable are changed from metres to centimetres, what happens to r?

  1. It stays the same Correct answer.
  2. It is multiplied by 100
  3. It becomes 0.009
  4. It is divided by 100

Explanation

The correlation coefficient is unit-free because it is calculated from standardised values. Changing the units of one variable rescales the values but does not change r.

Question 14Unit 3

A time series is smoothed using a 4-point moving average. Which statement is true?

  1. Even-numbered moving averages are centred automatically
  2. Each averaged value is centred between two periods and requires a further centring step Correct answer.
  3. A 4-point moving average removes the trend
  4. A 4-point moving average always produces a trend value for every data point

Explanation

With an even number of points, the moving average falls halfway between two time periods. A second centring step, such as a 2-point average of the moving averages, is needed to align values with actual periods.

Question 15Unit 3

Monthly sales data have a seasonal index of 1.25 for December. What does this index indicate?

  1. December sales are 1.25 units higher than average
  2. December sales are 125% below average
  3. December sales are typically 25% below the overall average level
  4. December sales are typically 25% above the overall average level Correct answer.

Explanation

A seasonal index compares a period's value to the average level. An index of 1.25 means the period is 25 percent above average, since 1.25 equals 125 percent of the average.

Question 16Unit 3

In deseasonalising time series data, a seasonal index of 0.8 is applied to a raw value of 200. What is the deseasonalised value?

  1. 160
  2. 199.2
  3. 250 Correct answer.
  4. 200.8

Explanation

To deseasonalise, divide the raw value by the seasonal index: 200 ÷ 0.8 = 250. This removes the seasonal effect and reveals the underlying level.

Question 17Unit 3

A data set has a correlation coefficient of r = -0.75. Which value is the slope of the least-squares line most likely to have?

  1. A positive value
  2. Exactly -0.75
  3. A negative value Correct answer.
  4. Exactly zero

Explanation

The slope of the least-squares line has the same sign as the correlation coefficient. Since r is negative, the slope must also be negative, though its size depends on the standard deviations.

Question 18Unit 3

A residual plot shows a clear U-shaped pattern. What does this suggest about the fitted model?

  1. The residuals have been miscalculated
  2. The correlation is exactly 1
  3. The linear model fits the data very well
  4. A linear model may not be appropriate for the data Correct answer.

Explanation

A residual plot should show random scatter if a linear model is appropriate. A systematic curved pattern such as a U-shape suggests the true relationship is nonlinear.

Question 19Unit 3

Two variables have a correlation coefficient of r = 0.6. A student claims that increasing the explanatory variable causes the response to increase. Which statement is the best response?

  1. The claim is valid because r is positive and greater than 0.5
  2. The claim is invalid because r is negative
  3. Correlation does not establish causation; the association may be due to another factor Correct answer.
  4. The claim is valid because r is close to 1

Explanation

A correlation describes association, not cause and effect. A third variable or coincidence could explain the relationship, so causation cannot be concluded from r alone.

Question 20Unit 3

A least-squares line is y = 30 - 0.5x, based on data where x ranges from 20 to 60. What is the predicted value of y when x = 40?

  1. 10 Correct answer.
  2. -10
  3. 20
  4. 50

Explanation

Substitute x = 40: y = 30 - 0.5(40) = 30 - 20 = 10. The slope is negative, so the prediction is below the intercept.

Question 21Unit 3

A time series has values 120, 130 and 140 in three consecutive periods. Using a 3-point moving average, what is the centred value for the second period?

  1. 130 Correct answer.
  2. 120
  3. 125
  4. 140

Explanation

A 3-point moving average for the middle period is the mean of the three values: (120 + 130 + 140) ÷ 3 = 130. For an odd window the average aligns directly with the middle period.

Question 22Unit 3

A least-squares regression line is fitted to data with correlation r = -0.8, standard deviation of x equal to 5 and standard deviation of y equal to 10. What is the slope of the line?

  1. -4
  2. -0.4
  3. -8
  4. -1.6 Correct answer.

Explanation

The slope of the least-squares line is b = r × (s_y ÷ s_x). Substituting gives b = -0.8 × (10 ÷ 5) = -0.8 × 2 = -1.6.

Question 23Unit 3

A least-squares line has slope 2.5 and passes through the point (4, 20). What is the intercept of the line?

  1. 30
  2. -10
  3. 10 Correct answer.
  4. 20

Explanation

Using y = a + bx with the given point: 20 = a + 2.5(4) = a + 10, so a = 10. The intercept is therefore 10.

Question 24Unit 3

A time series is modelled with a trend line and seasonal indices. A quarter has a raw value of 180 and a seasonal index of 1.2. The trend value for that quarter is 150. What is the residual after deseasonalising?

  1. -30
  2. 30
  3. 60
  4. 0 Correct answer.

Explanation

Deseasonalise by dividing the raw value by the index: 180 ÷ 1.2 = 150. The deseasonalised value equals the trend value, so the residual is 150 - 150 = 0.

Question 25Unit 3

For a set of data, the coefficient of determination is 0.81 and the correlation is negative. What is the value of the correlation coefficient?

  1. -0.81
  2. -0.9 Correct answer.
  3. 0.9
  4. -0.6561

Explanation

Since r² = 0.81, r is either 0.9 or -0.9. The sign of r matches the direction of the association, so a negative correlation gives r = -0.9.

Question 26Unit 3

A least-squares regression line is fitted to bivariate data, giving the equation y = 12.4 - 3.1x. What does the slope of this line indicate?

  1. For each one-unit increase in x, the predicted value of y decreases by 3.1 Correct answer.
  2. For each one-unit increase in x, the predicted value of y increases by 12.4
  3. For each one-unit increase in y, the predicted value of x decreases by 3.1
  4. The predicted value of y when x equals zero is -3.1

Explanation

In a least-squares line written as y = a + bx, the slope b is the predicted change in y for a one-unit increase in x. Here b = -3.1, so the predicted response falls by 3.1 units per unit rise in the explanatory variable.

Question 27Unit 3

A set of data has correlation coefficient r = 0.6. Which value is the slope of the least-squares line most likely to have?

  1. Zero
  2. Exactly equal to 0.6 in every case
  3. Negative
  4. Positive Correct answer.

Explanation

The slope of the least-squares line has the same sign as the correlation coefficient, because both are driven by the direction of the linear association. Since r = 0.6 is positive, the line must slope upward.

Question 28Unit 3

A time series of quarterly data is smoothed using a 4-point moving average. Which statement about this moving average is true?

  1. It removes the trend from the series
  2. It is centred by averaging two consecutive moving averages Correct answer.
  3. It cannot be used because the count is even
  4. It is centred automatically because the count is even

Explanation

An even-numbered moving average does not align with any single original data point, so a second averaging of consecutive moving averages is used to centre it. This two-stage process produces a centred moving average.

Question 29Unit 3

A seasonal index of 0.75 applies to the March quarter of a time series. What does this index indicate?

  1. March activity is typically about 25% above the average level
  2. March activity equals exactly 75 units in every year
  3. March activity is typically about 75% above the average level
  4. March activity is typically about 25% below the average level Correct answer.

Explanation

Seasonal indices are expressed relative to an average of 1. An index of 0.75 means the season is about three-quarters of the typical level, which is 25% below average.

Question 30Unit 3

A least-squares line is y = 8 - 2x. What is the residual for the data point (3, 5)?

  1. 3 Correct answer.
  2. -3
  3. 1
  4. -1

Explanation

The predicted value is 8 - 2(3) = 2. The residual is the observed value minus the predicted value, so 5 - 2 = 3.

Question 31Unit 3

A time series has values 200, 240, 220 and 260 in four consecutive periods. What is the first 3-point moving average?

  1. 220 Correct answer.
  2. 210
  3. 230
  4. 240

Explanation

A 3-point moving average at the centre of the first three periods is (200 + 240 + 220) / 3 = 660 / 3 = 220.

Question 32Unit 3

For a set of bivariate data, the coefficient of determination is 0.49. What is the value of the correlation coefficient?

  1. 0.49
  2. 0.7 or -0.7 Correct answer.
  3. 0.7 only
  4. 0.24 or -0.24

Explanation

The coefficient of determination equals r squared, so r is plus or minus the square root of 0.49, which is 0.7. The sign cannot be determined from r squared alone.

Question 33Unit 3

A time series is deseasonalised by dividing a raw value by its seasonal index. A raw value is 150 and its seasonal index is 1.2. What is the deseasonalised value?

  1. 125 Correct answer.
  2. 1.25
  3. 180
  4. 148.8

Explanation

Deseasonalising removes the seasonal effect, so the raw value is divided by the seasonal index: 150 / 1.2 = 125.

Question 34Unit 3

A least-squares line is y = 4 + 1.5x. The data used to fit it have x values ranging from 2 to 20. Which use of the line is an extrapolation?

  1. Predicting y when x = 12
  2. Predicting y when x = 5
  3. Predicting y when x = 18
  4. Predicting y when x = 35 Correct answer.

Explanation

Extrapolation means predicting outside the observed range of the explanatory variable. The observed range is 2 to 20, and x = 35 lies beyond it, so this prediction is an extrapolation.

Question 35Unit 3

A regression of y on x has correlation r = 0.8, standard deviation of x equal to 4 and standard deviation of y equal to 10. What is the slope of the least-squares line?

  1. 2 Correct answer.
  2. 5
  3. 3.2
  4. 0.32

Explanation

The slope equals r multiplied by the standard deviation of y divided by the standard deviation of x, so 0.8 x 10 / 4 = 2.

Question 36Unit 3

A least-squares line has slope -0.4 and passes through the point (5, 18). What is the y-intercept of the line?

  1. 20 Correct answer.
  2. -2
  3. 17.6
  4. 16

Explanation

Substituting into y = a + bx gives 18 = a - 0.4(5), so 18 = a - 2 and a = 20.

Question 37Unit 3

A residual plot from a least-squares fit shows residuals that fan out as the explanatory variable increases. Which conclusion is best supported?

  1. The relationship is perfectly linear
  2. The correlation coefficient must be exactly zero
  3. The assumption of constant variation about the line is doubtful Correct answer.
  4. The intercept has been calculated incorrectly

Explanation

A residual plot that widens as x increases indicates that the spread of residuals is not constant, which violates the usual assumption of constant variability. The line may still be useful, but predictions become less reliable at larger x.

Question 38Unit 3

In a least-squares regression, the coefficient of determination is 0.36. Which statement is correct?

  1. About 36% of the response values lie exactly on the line
  2. About 36% of the variation in the response is explained by the linear model Correct answer.
  3. The slope of the line is 0.36
  4. About 64% of the variation in the response is explained by the linear model

Explanation

The coefficient of determination gives the proportion of variation in the response variable accounted for by the least-squares line. A value of 0.36 means roughly 36% is explained, leaving 64% unexplained.

Question 39Unit 3

A quarterly time series has a trend value of 250 for a quarter whose seasonal index is 0.9. What is the seasonally adjusted forecast for that quarter?

  1. 249.1
  2. 225 Correct answer.
  3. 278
  4. 250.9

Explanation

A seasonal forecast multiplies the trend value by the seasonal index for that quarter: 250 x 0.9 = 225.

Question 40Unit 3

A least-squares line is y = 50 - 2x, based on data where x ranges from 10 to 30. What is the predicted value of y when x = 25?

  1. -2
  2. 0 Correct answer.
  3. 25
  4. 48

Explanation

Substituting x = 25 gives y = 50 - 2(25) = 50 - 50 = 0. Since 25 lies inside the observed range, this is interpolation.

Question 41Unit 3

A student finds that the correlation between hours of study and test score is r = 0.55. Which interpretation is appropriate?

  1. There is a strong negative linear association
  2. Studying causes scores to rise by 55%
  3. The least-squares line explains 55% of the variation in scores
  4. There is a moderate positive linear association between the two variables Correct answer.

Explanation

A correlation of 0.55 indicates a moderate positive linear association. Correlation measures the strength and direction of linear association, not causation or a percentage of explained variation.

Question 42Unit 3

A 4-point moving average is being centred. Two consecutive 4-point moving averages are 120 and 132. What is the centred moving average?

  1. 252
  2. 126 Correct answer.
  3. 120.5
  4. 6

Explanation

Centring an even moving average takes the mean of two consecutive values: (120 + 132) / 2 = 126.

Question 43Unit 3

In a set of bivariate data, the correlation coefficient is r = -0.85. If every value of the response variable is multiplied by 3, what is the new correlation coefficient?

  1. -0.28
  2. 0.85
  3. -2.55
  4. -0.85 Correct answer.

Explanation

Correlation is unaffected by changes of scale because it is based on standardised values. Multiplying all response values by a positive constant changes neither the strength nor the direction of the linear association.

Question 44Unit 3

A time series has a seasonal index of 1.4 for a quarter and a deseasonalised value of 90. What is the raw value for that quarter?

  1. 64
  2. 88.6
  3. 126 Correct answer.
  4. 91.4

Explanation

Since the deseasonalised value equals the raw value divided by the seasonal index, the raw value equals the deseasonalised value multiplied by the index: 90 x 1.4 = 126.

Question 45Unit 3

A scatterplot of two variables shows points tightly clustered around a straight line with positive slope. Which pair of values is most consistent with this plot?

  1. r = 0.50, coefficient of determination 0.25
  2. r = 0.95, coefficient of determination 0.90 Correct answer.
  3. r = 0.20, coefficient of determination 0.04
  4. r = -0.95, coefficient of determination 0.90

Explanation

A tight clustering about a positively sloped line indicates a strong positive linear association, so r should be close to 1. The coefficient of determination is then r squared, about 0.90.

Question 46Unit 3

A least-squares line has equation y = 15 + 0.8x. A data point has x = 20 and an observed y of 40. What is the residual for this point?

  1. 9 Correct answer.
  2. 71
  3. -9
  4. 31

Explanation

The predicted value is 15 + 0.8(20) = 15 + 16 = 31. The residual is observed minus predicted, so 40 - 31 = 9.

Question 47Unit 3

A least-squares line fitted to data has a negative slope. The correlation coefficient for the same data is r. Which statement must be true?

  1. r equals the slope
  2. r is greater than 1
  3. r is zero
  4. r is negative Correct answer.

Explanation

The slope and the correlation coefficient always share the same sign because both reflect the direction of the linear association. A negative slope therefore forces a negative correlation.

Question 48Unit 3

A least-squares regression is performed with x as the explanatory variable and y as the response. The means are x-bar = 12 and y-bar = 30, and the slope is 1.5. What is the y-intercept of the least-squares line?

  1. 12 Correct answer.
  2. 1.5
  3. 42
  4. 18

Explanation

Every least-squares line passes through the point of means, so 30 = a + 1.5(12) = a + 18, giving a = 12.

Question 49Unit 3

A data set has correlation r = -0.6, standard deviation of x equal to 5 and standard deviation of y equal to 8. What is the slope of the least-squares line predicting y from x?

  1. -3.75
  2. -0.96 Correct answer.
  3. 0.96
  4. -0.375

Explanation

The slope equals r times the standard deviation of y divided by the standard deviation of x: -0.6 x 8 / 5 = -0.96.

Question 50Unit 3

In a least-squares regression, the coefficient of determination is 0.64 and the correlation is positive. The standard deviation of x is 6 and the standard deviation of y is 15. What is the slope of the line?

  1. 2 Correct answer.
  2. 0.32
  3. 1.6
  4. 0.8

Explanation

Since r squared is 0.64 and r is positive, r = 0.8. The slope is r times the standard deviation of y divided by the standard deviation of x, giving 0.8 x 15 / 6 = 2.