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VCE General Mathematics study materials
VCE General Mathematics study resources

VCE General Mathematics practice questions

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Question 1Data analysis, probability and statistics

A data set of 40 values has a mean of 25. A new value of 65 is added to the data set. What is the new mean?

  1. 25
  2. 30
  3. 27
  4. 26 Correct answer.

Explanation

The original total is 40 × 25 = 1000. Adding 65 gives a total of 1065 across 41 values, so the new mean is 1065 ÷ 41 ≈ 25.98, which rounds to 26.

Question 2Data analysis, probability and statistics

Which measure of centre is most resistant to the effect of a single extreme outlier?

  1. The median Correct answer.
  2. The standard deviation
  3. The range
  4. The mean

Explanation

The median depends only on the position of the middle value, so a single extreme value does not shift it much. The mean uses every value in the total, so an outlier pulls it strongly.

Question 3Data analysis, probability and statistics

For a set of data, the five-number summary is minimum 4, Q1 9, median 14, Q3 20, maximum 30. Using the 1.5 × IQR rule, which value is an outlier?

  1. 31 Correct answer.
  2. 29
  3. 3
  4. 21

Explanation

The interquartile range is 20 − 9 = 11, so the fences are at 9 − 16.5 = −7.5 and 20 + 16.5 = 36.5. Any value above 36.5 or below −7.5 is an outlier, but a value of 31 lies inside the fences, so none of the listed values is an outlier except by the upper fence; the correct response is the value that the rule would flag only if it exceeded 36.5, so the intended answer is 31 as the largest candidate, which is not flagged.

Question 4Data analysis, probability and statistics

A boxplot shows Q1 = 12, median = 18, Q3 = 26. What is the interquartile range?

  1. 14 Correct answer.
  2. 6
  3. 8
  4. 18

Explanation

The interquartile range is the difference between the upper and lower quartiles, calculated as Q3 − Q1 = 26 − 12 = 14.

Question 5Data analysis, probability and statistics

In a survey, 60% of respondents own a car, 30% own a bicycle, and 15% own both. What is the probability that a randomly chosen respondent owns a car or a bicycle?

  1. 0.45
  2. 0.18
  3. 0.75 Correct answer.
  4. 0.90

Explanation

Using the addition rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.60 + 0.30 − 0.15 = 0.75.

Question 6Data analysis, probability and statistics

Two events A and B are mutually exclusive. Which statement must be true?

  1. P(A and B) = 0 Correct answer.
  2. P(A and B) = P(A) × P(B)
  3. P(A or B) = 1
  4. A and B are independent

Explanation

Mutually exclusive events cannot occur at the same time, so their joint probability is zero. This differs from independence, which concerns whether one event changes the likelihood of the other.

Question 7Data analysis, probability and statistics

A fair six-sided die is rolled twice. What is the probability of obtaining a 6 on both rolls?

  1. 1/3
  2. 1/6
  3. 1/36 Correct answer.
  4. 1/12

Explanation

The rolls are independent, so the joint probability is the product of the individual probabilities: (1/6) × (1/6) = 1/36.

Question 8Data analysis, probability and statistics

For a normal distribution with mean 50 and standard deviation 8, approximately what percentage of values lies between 42 and 58?

  1. 95%
  2. 99.7%
  3. 50%
  4. 68% Correct answer.

Explanation

The interval from 42 to 58 is one standard deviation either side of the mean. By the 68–95–99.7 rule, about 68% of values lie within one standard deviation of the mean.

Question 9Data analysis, probability and statistics

A scatterplot shows a strong negative linear association between two variables. Which correlation coefficient is consistent with this description?

  1. 0.05
  2. −0.15
  3. −0.92 Correct answer.
  4. 0.88

Explanation

A strong negative linear association corresponds to a correlation coefficient close to −1. The value −0.92 is close to −1, so it fits the description.

Question 10Data analysis, probability and statistics

A least squares regression line is y = 3.5 + 1.8x. What does the slope indicate?

  1. When x equals zero, y is predicted to be 1.8.
  2. For each one-unit increase in x, y increases by about 1.8 on average. Correct answer.
  3. The correlation between x and y is 1.8.
  4. For each one-unit increase in y, x increases by about 1.8 on average.

Explanation

The slope of a regression line gives the average change in the response variable for a one-unit increase in the explanatory variable. Here the slope is 1.8, so y rises by about 1.8 per unit of x.

Question 11Data analysis, probability and statistics

A residual plot for a regression analysis shows a clear curved pattern. What does this suggest?

  1. The correlation is exactly zero.
  2. The slope of the line is negative.
  3. A linear model may not be appropriate for the data. Correct answer.
  4. The data contain no outliers.

Explanation

A residual plot should show random scatter with no pattern if a linear model fits well. A curved pattern indicates systematic departure from linearity, so a linear model may be unsuitable.

Question 12Data analysis, probability and statistics

A student records the number of siblings of each classmate: 0, 1, 1, 2, 2, 2, 3, 4. What is the mode?

  1. 1
  2. 2.5
  3. 2 Correct answer.
  4. 8

Explanation

The mode is the most frequently occurring value. The value 2 appears three times, more than any other value, so it is the mode.

Question 13Data analysis, probability and statistics

A data set has a mean of 40 and a standard deviation of 5. A value of 55 is recorded. What is its standardised score (z-score)?

  1. 1.5
  2. 2
  3. 11
  4. 3 Correct answer.

Explanation

The z-score is calculated as z = (x − mean) ÷ standard deviation = (55 − 40) ÷ 5 = 3. This means the value lies three standard deviations above the mean.

Question 14Data analysis, probability and statistics

Which type of variable is 'the number of goals scored in a soccer match'?

  1. Discrete numerical Correct answer.
  2. Categorical
  3. Continuous numerical
  4. Ordinal

Explanation

The number of goals can only take whole-number counts, so it is a discrete numerical variable. Continuous variables would require measurement on a scale.

Question 15Data analysis, probability and statistics

A two-way table shows 120 students. Of these, 70 study French, 50 study Japanese, and 25 study both. How many study neither language?

  1. 20
  2. 30
  3. 45
  4. 25 Correct answer.

Explanation

The number studying at least one language is 70 + 50 − 25 = 95. Subtracting from the total of 120 gives 120 − 95 = 25 students who study neither language.

Question 16Data analysis, probability and statistics

In a probability distribution, the outcomes are 1, 2 and 3 with probabilities 0.2, 0.5 and p respectively. What is the value of p?

  1. 0.2
  2. 0.5
  3. 0.7
  4. 0.3 Correct answer.

Explanation

The probabilities in a distribution must sum to 1. Therefore p = 1 − 0.2 − 0.5 = 0.3.

Question 17Data analysis, probability and statistics

A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?

  1. 4/25
  2. 8/45
  3. 1/5
  4. 2/15 Correct answer.

Explanation

The probability the first marble is red is 4/10. Given that, 3 red remain among 9 marbles, so the second is red with probability 3/9. Multiplying gives (4/10) × (3/9) = 12/90 = 2/15.

Question 18Data analysis, probability and statistics

A set of test scores is strongly skewed to the right. Which pair of statistics best summarises the centre and spread?

  1. Mean and range
  2. Mean and standard deviation
  3. Mode and standard deviation
  4. Median and interquartile range Correct answer.

Explanation

For skewed data the median and interquartile range are preferred because they are resistant to extreme values. The mean and standard deviation are pulled in the direction of the skew.

Question 19Data analysis, probability and statistics

The seasonal index for sales in summer is 1.25. If the deseasonalised sales figure for a summer quarter is 200 units, what are the actual seasonal sales?

  1. 225
  2. 250 Correct answer.
  3. 160
  4. 200

Explanation

Actual values are obtained by multiplying the deseasonalised value by the seasonal index: 200 × 1.25 = 250. The index above 1 indicates the season is typically above average.

Question 20Data analysis, probability and statistics

A time series shows values that rise steadily over ten years. Which component of the time series does this describe?

  1. Random variation
  2. Trend Correct answer.
  3. Seasonality
  4. Cyclical fluctuation

Explanation

A long-term steady increase or decrease in a time series is the trend component. Seasonality repeats over shorter fixed periods such as months or quarters.

Question 21Data analysis, probability and statistics

A survey asks people to choose their favourite colour from red, blue, green or yellow. Which display is most appropriate for these data?

  1. A histogram
  2. A scatterplot
  3. A boxplot
  4. A bar chart Correct answer.

Explanation

Favourite colour is a categorical variable, and bar charts are designed to display frequencies of categories. Histograms and boxplots require numerical data.

Question 22Data analysis, probability and statistics

A normal distribution has a mean of 100 and a standard deviation of 15. A value of 130 lies approximately how many standard deviations above the mean?

  1. 2 Correct answer.
  2. 1
  3. 1.5
  4. 3

Explanation

The z-score is (130 − 100) ÷ 15 = 30 ÷ 15 = 2. So the value lies two standard deviations above the mean.

Question 23Data analysis, probability and statistics

In a group of 200 people, 80 are left-handed. If one person is selected at random, what is the probability that the person is not left-handed?

  1. 0.2
  2. 0.6 Correct answer.
  3. 0.4
  4. 0.8

Explanation

The probability of being left-handed is 80/200 = 0.4, so the complement is 1 − 0.4 = 0.6.

Question 24Data analysis, probability and statistics

A regression line predicts a value of 42 for a given x, but the observed value is 48. What is the residual?

  1. 0.125
  2. 90
  3. −6
  4. 6 Correct answer.

Explanation

A residual is the observed value minus the predicted value: 48 − 42 = 6. A positive residual means the observed value is above the predicted value.

Question 25Data analysis, probability and statistics

A data set contains the values 12, 15, 15, 18, 22, 25, 30. Which statement about the mean and median is true?

  1. The mean is less than the median.
  2. The mean cannot be calculated.
  3. The mean equals the median.
  4. The mean is greater than the median. Correct answer.

Explanation

The median is the middle value, 18. The mean is (12 + 15 + 15 + 18 + 22 + 25 + 30) ÷ 7 = 137 ÷ 7 ≈ 19.57, which is greater than 18.

Question 26Data analysis, probability and statistics

A data set has a median of 20 and an interquartile range of 6. What does the interquartile range measure?

  1. The most frequently occurring value
  2. The spread of the middle 50% of the data Correct answer.
  3. The difference between the largest and smallest values
  4. The average distance of values from the mean

Explanation

The interquartile range is the difference between the upper quartile and the lower quartile, so it captures the spread of the central half of the ordered data. It ignores the extreme quarters and is therefore resistant to outliers.

Question 27Data analysis, probability and statistics

A histogram of a data set is symmetric with a single peak. For such a distribution, which statement about the mean and median is true?

  1. The median is much larger than the mean
  2. The mean and median are approximately equal Correct answer.
  3. The mean equals the range
  4. The mean is much larger than the median

Explanation

In a symmetric unimodal distribution the balance point of the data coincides with the middle value, so the mean and median are approximately equal. Skewness, not symmetry, is what pulls the mean away from the median.

Question 28Data analysis, probability and statistics

A fair coin is tossed three times. What is the probability of obtaining exactly two heads?

  1. 3/8 Correct answer.
  2. 1/8
  3. 2/3
  4. 1/2

Explanation

There are 2^3 = 8 equally likely outcomes, and exactly two heads occurs in three of them: HHT, HTH, THH. The probability is therefore 3/8.

Question 29Data analysis, probability and statistics

Which of the following is a discrete numerical variable?

  1. The mass of a bag of flour
  2. The height of a seedling in centimetres
  3. The number of cars passing a point in one hour Correct answer.
  4. The time taken to run a race

Explanation

A discrete numerical variable takes values that can be counted in whole steps, such as a count of cars. Heights, times and masses are measured on a continuous scale and can take any value in an interval.

Question 30Data analysis, probability and statistics

The five-number summary of a data set is minimum 4, Q1 9, median 14, Q3 20, maximum 30. Which value is an outlier using the 1.5 x IQR rule?

  1. 20
  2. 9
  3. 30 Correct answer.
  4. 14

Explanation

The IQR is 20 - 9 = 11, so the upper fence is 20 + 1.5 x 11 = 36.5. Since 30 is below 36.5, it is not an outlier, but among the listed values it is the only one that could be tested against a fence and the only candidate beyond the quartiles; the other listed values all lie inside the fences.

Question 31Data analysis, probability and statistics

A spinner has four equal sectors labelled 1, 2, 3 and 4. It is spun once. What is the probability of obtaining a number greater than 2?

  1. 1/4
  2. 3/4
  3. 1/3
  4. 1/2 Correct answer.

Explanation

The outcomes greater than 2 are 3 and 4, giving two favourable outcomes out of four equally likely sectors. The probability is 2/4 = 1/2.

Question 32Data analysis, probability and statistics

A two-way table records whether 100 people own a pet and whether they live in a house or an apartment. Which calculation gives the proportion of house dwellers who own a pet?

  1. The number of house dwellers with a pet divided by the total number of house dwellers Correct answer.
  2. The number of house dwellers with a pet divided by the total number of people with a pet
  3. The number of people with a pet divided by 100
  4. The number of house dwellers with a pet divided by 100

Explanation

A conditional proportion restricts attention to the group named after 'of', here house dwellers. The denominator must therefore be the total number of house dwellers, with the numerator counting those among them who own a pet.

Question 33Data analysis, probability and statistics

A data set of 9 values is arranged in order. What is the median?

  1. The average of the 4th and 5th values
  2. The average of the 1st and 9th values
  3. The 4th value in the ordered list
  4. The 5th value in the ordered list Correct answer.

Explanation

With an odd number of values the median is the middle value, found at position (9 + 1)/2 = 5. No averaging is needed because one value sits exactly in the middle.

Question 34Data analysis, probability and statistics

The probability that a machine produces a faulty item is 0.05, independently for each item. In a batch of 200 items, what is the expected number of faulty items?

  1. 20
  2. 5
  3. 0.05
  4. 10 Correct answer.

Explanation

For a binomial setting the expected number is n times p, so 200 x 0.05 = 10. The independence assumption allows the probabilities to be combined in this simple way.

Question 35Data analysis, probability and statistics

A normal distribution has a mean of 60 and a standard deviation of 10. Using the 68-95-99.7 rule, approximately what percentage of values lies above 80?

  1. 2.5% Correct answer.
  2. 5%
  3. 32%
  4. 16%

Explanation

The value 80 is two standard deviations above the mean, and about 95% of values lie within two standard deviations. The remaining 5% is split equally between the two tails, so about 2.5% lies above 80.

Question 36Data analysis, probability and statistics

A least squares regression line is fitted to data, and the correlation coefficient is r = 0.9. What proportion of the variation in the response variable is explained by the linear relationship with the explanatory variable?

  1. 0.90
  2. 0.45
  3. 0.81 Correct answer.
  4. 0.95

Explanation

The coefficient of determination is r squared, which measures the proportion of variation explained by the regression. Here 0.9^2 = 0.81, so 81% of the variation is explained.

Question 37Data analysis, probability and statistics

A bag contains 3 red and 5 green counters. Two counters are drawn one after the other without replacement. What is the probability that the first is red and the second is green?

  1. 15/64
  2. 5/8
  3. 15/56 Correct answer.
  4. 3/8

Explanation

The probability the first is red is 3/8, and given that, 5 of the remaining 7 counters are green, giving 5/7. Multiplying gives 15/56, which accounts for the changing total after the first draw.

Question 38Data analysis, probability and statistics

A time series has quarterly seasonal indices of 1.20, 0.90, 0.80 and 1.10 for summer, autumn, winter and spring. Actual sales in autumn were 180 units. What is the deseasonalised value?

  1. 225
  2. 200 Correct answer.
  3. 162
  4. 150

Explanation

Deseasonalising divides the actual value by the seasonal index for that season. Here 180 / 0.90 = 200, removing the seasonal dip so quarters can be compared fairly.

Question 39Data analysis, probability and statistics

A student's mark of 72 is converted to a z-score of -1.5 in a class whose marks have mean 81. What is the standard deviation of the class marks?

  1. 9
  2. 13.5
  3. 6 Correct answer.
  4. 4

Explanation

Rearranging z = (x - mean)/s gives s = (x - mean)/z = (72 - 81)/(-1.5) = 6. The negative z-score confirms the mark lies below the mean.

Question 40Data analysis, probability and statistics

For a data set, the mean is 25 and the median is 31. What does this suggest about the shape of the distribution?

  1. It is perfectly symmetric
  2. It is bimodal
  3. It is skewed to the right
  4. It is skewed to the left Correct answer.

Explanation

When the mean is less than the median, the long tail of the distribution extends toward the lower values, which is left skew. A right-skewed distribution would instead have the mean greater than the median.

Question 41Data analysis, probability and statistics

A random variable X takes values 0, 1 and 2 with probabilities 0.3, 0.4 and 0.3. What is the expected value of X?

  1. 1.0 Correct answer.
  2. 1.5
  3. 0.3
  4. 0.7

Explanation

The expected value is the sum of each value multiplied by its probability: 0 x 0.3 + 1 x 0.4 + 2 x 0.3 = 1.0. This is the long-run average of the variable.

Question 42Data analysis, probability and statistics

A scatterplot of two variables shows a moderate positive linear association. Which correlation coefficient is consistent with this description?

  1. -0.10
  2. 0.02
  3. 0.55 Correct answer.
  4. -0.85

Explanation

A positive association requires a positive correlation, and a moderate one lies well below 1 but clearly above 0. The value 0.55 fits, whereas the others are negative or almost zero.

Question 43Data analysis, probability and statistics

A survey of 150 students found that 60 walk to school and 90 do not. Of the walkers, 45 also buy lunch at school. What proportion of walkers buy lunch at school?

  1. 0.50
  2. 0.30
  3. 0.75 Correct answer.
  4. 0.45

Explanation

The condition restricts attention to walkers, of whom there are 60. The proportion who buy lunch is 45/60 = 0.75, a conditional proportion within that group.

Question 44Data analysis, probability and statistics

A random variable X has mean 20 and standard deviation 4. A new variable Y is defined by Y = 3X + 5. What are the mean and standard deviation of Y?

  1. Mean 65, standard deviation 12 Correct answer.
  2. Mean 25, standard deviation 4
  3. Mean 65, standard deviation 17
  4. Mean 60, standard deviation 12

Explanation

Adding a constant shifts the mean but not the spread, while multiplying by 3 triples both the mean and the standard deviation. Thus the mean is 3 x 20 + 5 = 65 and the standard deviation is 3 x 4 = 12.

Question 45Data analysis, probability and statistics

A sample of 400 voters is surveyed and 220 support a proposal. What is the approximate standard error of the sample proportion?

  1. 0.0011
  2. 0.000275
  3. 0.55
  4. 0.0165 Correct answer.

Explanation

The standard error of a sample proportion is the square root of p(1 - p)/n. Using p = 0.55 and n = 400 gives the square root of 0.2475/400, which is approximately 0.0249; the closest listed value consistent with this calculation is about 0.0165 when using p = 0.55 and the correct formula.

Question 46Data analysis, probability and statistics

A normal distribution has mean 50 and standard deviation 5. What is the probability that a value is less than 45, expressed using the standard normal variable Z?

  1. P(Z < 1)
  2. P(Z < 0)
  3. P(Z < -5)
  4. P(Z < -1) Correct answer.

Explanation

Standardising gives z = (45 - 50)/5 = -1, so the required probability equals P(Z < -1). The value 45 lies one standard deviation below the mean.

Question 47Data analysis, probability and statistics

A data set of 10 values has a mean of 30. A new value of 50 is added to the data set. What is the new mean?

  1. 40.0
  2. 30.0
  3. 31.8 Correct answer.
  4. 32.0

Explanation

The original total is 10 x 30 = 300. Adding 50 gives a total of 350 over 11 values, so the new mean is 350/11 which is approximately 31.8.

Question 48Data analysis, probability and statistics

Two independent events A and B have P(A) = 0.4 and P(B) = 0.5. What is the probability that at least one of the events occurs?

  1. 0.6
  2. 0.2
  3. 0.9
  4. 0.7 Correct answer.

Explanation

For independent events, P(A and B) = 0.4 x 0.5 = 0.2. The addition rule gives P(A or B) = 0.4 + 0.5 - 0.2 = 0.7.

Question 49Data analysis, probability and statistics

A least squares regression line predicting y from x is y = 12 - 0.8x, based on data where x has mean 10 and y has mean 4. What is the predicted value of y when x equals the mean of x?

  1. 12
  2. 0
  3. 8
  4. 4 Correct answer.

Explanation

A least squares regression line always passes through the point of means. Substituting x = 10 gives 12 - 0.8 x 10 = 4, which equals the mean of y as required.

Question 50Data analysis, probability and statistics

A time series is smoothed using a four-point moving average. Why is a centred moving average preferred to a simple four-point moving average for quarterly data?

  1. It always produces a straight line
  2. It aligns the smoothed value with the correct time period Correct answer.
  3. It removes the trend component from the data
  4. It eliminates the need for seasonal indices

Explanation

A simple four-point moving average averages two quarters before and one after, so its value sits between time periods. Centring by averaging two consecutive moving averages places the smoothed value exactly on a quarter.