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VCE Foundation Mathematics study materials
VCE Foundation Mathematics study resources

VCE Foundation Mathematics practice questions

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Question 1Data analysis, probability and statistics

A café records the number of coffees sold each day for one week: 42, 38, 51, 47, 44, 60, 38. What is the mode of this data set?

  1. 44
  2. 38 Correct answer.
  3. 42
  4. 47

Explanation

The mode is the value that occurs most often. In this list the number 38 appears twice while every other value appears only once, so the mode is 38.

Question 2Data analysis, probability and statistics

The heights in centimetres of five seedlings are 12, 15, 18, 21, 24. What is the median height?

  1. 21
  2. 15
  3. 18 Correct answer.
  4. 24

Explanation

When the values are arranged in order, the median is the middle value. With five ordered values the third one is the middle, and here that value is 18.

Question 3Data analysis, probability and statistics

A survey records eye colour as blue, brown, green or hazel. Which type of data is this?

  1. Categorical nominal Correct answer.
  2. Numerical discrete
  3. Numerical continuous
  4. Categorical ordinal

Explanation

Eye colour is recorded as a label or name rather than a number, so the data is categorical. The categories have no natural order or ranking, which makes them nominal rather than ordinal.

Question 4Data analysis, probability and statistics

A spinner has 8 equal sectors, of which 3 are red. If the spinner is spun once, what is the probability that it lands on red?

  1. 1/8
  2. 5/8
  3. 3/5
  4. 3/8 Correct answer.

Explanation

For equally likely outcomes the probability equals the number of favourable outcomes divided by the total number of outcomes. There are 3 red sectors out of 8 equal sectors, so the probability is 3/8.

Question 5Data analysis, probability and statistics

A data set of 40 values is displayed in a frequency table. What is the mean of the data set?

  1. The difference between the largest and smallest values
  2. The middle value when the data is ordered
  3. The sum of all values divided by 40 Correct answer.
  4. The most frequently occurring value

Explanation

The mean is calculated by adding all the data values and dividing by the number of values. With 40 values, this means dividing the total sum by 40.

Question 6Data analysis, probability and statistics

A fair six-sided die is rolled once. What is the probability of rolling a number greater than 4?

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

Explanation

The outcomes greater than 4 are 5 and 6, giving 2 favourable outcomes out of 6 equally likely outcomes. Simplifying 2/6 gives 1/3.

Question 7Data analysis, probability and statistics

Which measure of spread is found by subtracting the smallest value from the largest value in a data set?

  1. Standard deviation
  2. Range Correct answer.
  3. Interquartile range
  4. Median

Explanation

The range is defined as the largest value minus the smallest value and gives a quick indication of total spread. It depends only on the two extreme values.

Question 8Data analysis, probability and statistics

A bag contains 5 red and 7 blue marbles. One marble is drawn at random. What is the probability that it is blue?

  1. 5/7
  2. 5/12
  3. 7/5
  4. 7/12 Correct answer.

Explanation

The total number of marbles is 5 + 7 = 12, and 7 of these are blue. Dividing the favourable count by the total gives 7/12.

Question 9Data analysis, probability and statistics

The daily rainfall in millimetres over six days is 0, 2, 4, 6, 8, 10. What is the mean daily rainfall?

  1. 5 mm Correct answer.
  2. 10 mm
  3. 6 mm
  4. 4 mm

Explanation

Adding the six values gives 30, and dividing by the six days gives 5. The mean is therefore 5 mm, which balances the values around it.

Question 10Data analysis, probability and statistics

A data set has a lower quartile of 12 and an upper quartile of 28. What is the interquartile range?

  1. 28
  2. 20
  3. 40
  4. 16 Correct answer.

Explanation

The interquartile range is the upper quartile minus the lower quartile. Subtracting 12 from 28 gives 16, which describes the spread of the middle half of the data.

Question 11Data analysis, probability and statistics

A student's scores on four tests are 60, 70, 80 and 90. A fifth test score is added and the mean of all five scores becomes 78. What was the fifth score?

  1. 90 Correct answer.
  2. 85
  3. 95
  4. 80

Explanation

The total of five scores must be 5 × 78 = 390. The first four scores total 300, so the fifth score is 390 − 300 = 90.

Question 12Data analysis, probability and statistics

Two fair coins are tossed. What is the probability that both coins land on heads?

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

Explanation

The four equally likely outcomes are HH, HT, TH and TT. Only one of these outcomes has two heads, so the probability is 1/4.

Question 13Data analysis, probability and statistics

A histogram of a data set is skewed to the right. Which statement about its measures of centre is generally true?

  1. The mean equals the median
  2. The mean equals the mode
  3. The mean is less than the median
  4. The mean is greater than the median Correct answer.

Explanation

Right skew means a long tail of unusually large values pulls the mean upward. The median is resistant to these extreme values, so the mean generally exceeds the median.

Question 14Data analysis, probability and statistics

In a class of 25 students, 10 study French, 12 study Japanese and 5 study both languages. How many students study neither language?

  1. 7
  2. 3 Correct answer.
  3. 8
  4. 2

Explanation

By the addition rule the number studying at least one language is 10 + 12 − 5 = 17. Subtracting from the class total gives 25 − 17 = 8 students who study neither.

Question 15Data analysis, probability and statistics

A box contains 4 red and 6 green counters. One counter is drawn and not replaced, then a second counter is drawn. What is the probability that both counters are red?

  1. 1/6
  2. 2/15 Correct answer.
  3. 4/10
  4. 2/5

Explanation

The first draw is red with probability 4/10. After a red counter is removed, 3 of the remaining 9 are red, so the second draw is red with probability 3/9. Multiplying gives 12/90 = 2/15.

Question 16Data analysis, probability and statistics

A set of test scores has a mean of 65 and a standard deviation of 5. A student scores 75. How many standard deviations above the mean is this score?

  1. 2 Correct answer.
  2. 1
  3. 10
  4. 1.5

Explanation

The number of standard deviations is found by dividing the difference from the mean by the standard deviation. Here (75 − 65) ÷ 5 = 10 ÷ 5 = 2.

Question 17Data analysis, probability and statistics

A survey asks people to choose their favourite season from summer, autumn, winter and spring. Which display is most appropriate for showing the proportion choosing each season?

  1. A pie chart Correct answer.
  2. A histogram
  3. A line graph
  4. A scatterplot

Explanation

A pie chart divides a circle into sectors proportional to each category's share of the whole. This suits categorical data where the parts add up to a single total.

Question 18Data analysis, probability and statistics

The probability that a train is late on any given day is 0.2, independently of other days. What is the probability that the train is on time on two consecutive days?

  1. 0.64 Correct answer.
  2. 0.16
  3. 0.8
  4. 0.04

Explanation

The probability of being on time on one day is 1 − 0.2 = 0.8. Because the days are independent, multiply: 0.8 × 0.8 = 0.64.

Question 19Data analysis, probability and statistics

A data set of 9 values is arranged in order. The median is removed and the remaining 8 values are used to find the quartiles. Which value is the lower quartile?

  1. The 2nd value
  2. The 3rd value
  3. The 2.5th value Correct answer.
  4. The 4th value

Explanation

With 8 values below the median, the lower quartile is the median of those 8 values. The median of an even count is the average of the 4th and 5th values, positioned at 2.5 in the lower half.

Question 20Data analysis, probability and statistics

In a two-way table, 60% of students are female, and 30% of all students play a sport. If gender and playing a sport are independent, what percentage of female students play a sport?

  1. 60%
  2. 18%
  3. 50%
  4. 30% Correct answer.

Explanation

Independence means the conditional probability equals the overall probability. Since 30% of all students play a sport, 30% of female students also play a sport.

Question 21Data analysis, probability and statistics

A data set has a mean of 50 and a standard deviation of 0. A student claims the values must all be different. Which statement is correct?

  1. The standard deviation cannot be zero
  2. The values must all equal 50 Correct answer.
  3. The values are evenly spaced around 50
  4. The values are all zero

Explanation

Standard deviation measures the typical distance of values from the mean. A standard deviation of zero means no value deviates from the mean, so every value equals 50.

Question 22Data analysis, probability and statistics

A box plot shows a lower whisker at 5, lower quartile 12, median 20, upper quartile 30 and upper whisker 45. Which interval contains exactly 25% of the data?

  1. 30 to 45
  2. 20 to 30 Correct answer.
  3. 12 to 20
  4. 5 to 12

Explanation

Each section between consecutive quartiles contains about 25% of the data. The interval from the lower quartile 20 to the upper quartile 30 is not the correct quartile boundary pair; the section from the median 20 to the upper quartile 30 holds the third quarter of the data.

Question 23Data analysis, probability and statistics

Two events A and B satisfy P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. What is P(A or B)?

  1. 0.7 Correct answer.
  2. 1.1
  3. 0.9
  4. 0.5

Explanation

The addition rule states P(A or B) = P(A) + P(B) − P(A and B). Substituting gives 0.5 + 0.4 − 0.2 = 0.7, which correctly avoids double-counting the overlap.

Question 24Data analysis, probability and statistics

A data set is transformed by adding 10 to every value. How do the mean and standard deviation change?

  1. The mean increases by 10 and the standard deviation is unchanged Correct answer.
  2. The mean is unchanged and the standard deviation increases by 10
  3. Both are unchanged
  4. Both increase by 10

Explanation

Adding a constant shifts every value and the mean by that constant, so the mean rises by 10. The distances between values stay the same, so the spread and the standard deviation are unchanged.

Question 25Data analysis, probability and statistics

A test has 20 multiple-choice questions, each with 4 options and exactly one correct answer. A student guesses every answer. What is the expected number of correct answers?

  1. 8
  2. 10
  3. 4
  4. 5 Correct answer.

Explanation

Each guess is correct with probability 1/4. The expected number of successes is the number of trials multiplied by that probability, giving 20 × 1/4 = 5.

Question 26Data analysis, probability and statistics

A data set contains the values 3, 7, 7, 9 and 14. What is the mode?

  1. 7 Correct answer.
  2. 3
  3. 9
  4. 8

Explanation

The mode is the value that occurs most often in a data set. In this list the number 7 appears twice while every other value appears once, so 7 is the mode.

Question 27Data analysis, probability and statistics

A spinner has 10 equal sectors numbered 1 to 10. What is the probability that the spinner lands on a multiple of 3?

  1. 4/10
  2. 3/10 Correct answer.
  3. 1/3
  4. 2/10

Explanation

The multiples of 3 from 1 to 10 are 3, 6 and 9, giving three favourable outcomes. With ten equally likely sectors the probability is 3 out of 10, written 3/10.

Question 28Data analysis, probability and statistics

The ages, in years, of five children are 4, 6, 7, 9 and 14. What is the median age?

  1. 8
  2. 9
  3. 6
  4. 7 Correct answer.

Explanation

The median is the middle value when the data are arranged in order. These five values are already ordered, so the third value, 7, is the median.

Question 29Data analysis, probability and statistics

A survey records the eye colour of 200 students as blue, brown, green or hazel. Which type of data is being collected?

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

Explanation

Eye colour places each student into a named group rather than assigning a number, so the variable is categorical. Numerical data would require a measured or counted quantity such as height or number of siblings.

Question 30Data analysis, probability and statistics

A fair coin is tossed and a fair six-sided die is rolled. What is the probability of getting a head and a 5?

  1. 1/12 Correct answer.
  2. 1/2
  3. 1/8
  4. 1/6

Explanation

The coin and the die are independent, so the combined probability is the product of the separate probabilities. That gives 1/2 multiplied by 1/6, which equals 1/12.

Question 31Data analysis, probability and statistics

The number of pets owned by each of 20 students is recorded. Which graph is most suitable for displaying the frequency of each possible number of pets?

  1. A line graph of a trend over time
  2. A pie chart of a single total
  3. A scatterplot of two numerical variables
  4. A bar chart Correct answer.

Explanation

The number of pets is a discrete numerical variable with a small set of possible values, so a bar chart can show the frequency of each value clearly. Bar charts suit counts of categories or discrete values.

Question 32Data analysis, probability and statistics

A data set has a mean of 40 and a median of 52. Which statement best describes the shape of the distribution?

  1. It is perfectly symmetric
  2. It has no variation
  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, a long tail of unusually small values pulls the mean downwards, which indicates left skew. In a right-skewed distribution the mean would be greater than the median.

Question 33Data analysis, probability and statistics

A bag contains 8 counters numbered 1 to 8. One counter is drawn at random. What is the probability of drawing a number less than 3?

  1. 5/8
  2. 1/8
  3. 3/8
  4. 2/8 Correct answer.

Explanation

The numbers less than 3 are 1 and 2, giving two favourable outcomes out of eight equally likely counters. The probability is therefore 2/8.

Question 34Data analysis, probability and statistics

The values 12, 15, 15, 18, 20, 22, 24, 26, 30 are arranged in order. What is the median?

  1. 18
  2. 20 Correct answer.
  3. 21
  4. 22

Explanation

There are nine values, so the median is the fifth value in the ordered list. Counting to the fifth value gives 20.

Question 35Data analysis, probability and statistics

A data set of 40 values has a lower quartile of 18 and an upper quartile of 34. A value of 50 is recorded. Using the rule that a value is an outlier if it lies more than 1.5 times the interquartile range beyond a quartile, is 50 an outlier?

  1. Yes, because it lies above the upper fence of 58
  2. Yes, because it exceeds 58 is false, so it is not an outlier
  3. No, because it is within one standard deviation
  4. No, because it is below the upper fence Correct answer.

Explanation

The interquartile range is 34 minus 18, which equals 16, and 1.5 times this is 24. The upper fence is 34 plus 24, which equals 58, and 50 is below 58, so it is not an outlier.

Question 36Data analysis, probability and statistics

Two events A and B are mutually exclusive with P(A) = 0.35 and P(B) = 0.25. What is the probability that neither A nor B occurs?

  1. 0.40 Correct answer.
  2. 0.60
  3. 0.10
  4. 0.0875

Explanation

For mutually exclusive events, P(A or B) equals P(A) plus P(B), which is 0.60. The probability that neither occurs is 1 minus 0.60, giving 0.40.

Question 37Data analysis, probability and statistics

A set of five values has a mean of 12. Four of the values are 8, 10, 14 and 15. What is the fifth value?

  1. 11
  2. 13 Correct answer.
  3. 16
  4. 12

Explanation

The total of all five values is the mean multiplied by the count, which is 12 times 5, or 60. The four known values sum to 47, so the fifth value is 60 minus 47, which is 13.

Question 38Data analysis, probability and statistics

A data set has a mean of 30 and a standard deviation of 4. A value of 22 is recorded. What is the standardised score (z-score) of this value?

  1. -0.5
  2. -8
  3. -2 Correct answer.
  4. 2

Explanation

The z-score is the deviation from the mean divided by the standard deviation. Here 22 minus 30 equals negative 8, and dividing by 4 gives negative 2.

Question 39Data analysis, probability and statistics

A random sample of 50 students has a mean height of 165 cm. Which statement about the population mean height is most appropriate?

  1. It cannot be estimated from a single sample
  2. It is exactly 165 cm because the sample mean is unbiased
  3. It must be greater than 165 cm because samples underestimate the mean
  4. It is likely to be near 165 cm, but a different sample would give a different sample mean Correct answer.

Explanation

A sample mean is an estimate of the population mean and varies from sample to sample because of sampling variability. A single sample of 50 gives a reasonable estimate but not an exact value.

Question 40Data analysis, probability and statistics

A two-way table shows that 30 students play an instrument and 20 do not, and that 18 of the instrument players are female. If a student is chosen at random from all 50 students, what is the probability of choosing a female who plays an instrument?

  1. 18/30
  2. 18/50 Correct answer.
  3. 30/50
  4. 18/20

Explanation

The probability of a combined event is the count in that cell divided by the total number of students. There are 18 females who play an instrument out of 50 students, giving 18/50.

Question 41Data analysis, probability and statistics

A data set of 6 values is 4, 8, 9, 11, 13 and 15. What is the mean absolute deviation from the mean?

  1. 6
  2. 2.5
  3. 3 Correct answer.
  4. 9

Explanation

The mean is 60 divided by 6, which is 10. The absolute deviations are 6, 2, 1, 1, 3 and 5, which sum to 18, and dividing by 6 gives 3.

Question 42Data analysis, probability and statistics

A box contains 3 white and 5 black balls. Two balls are drawn without replacement. What is the probability that both are white?

  1. 1/4
  2. 3/28 Correct answer.
  3. 9/64
  4. 3/8

Explanation

The probability the first ball is white is 3/8, and then 2 white balls remain among 7. Multiplying 3/8 by 2/7 gives 6/56, which simplifies to 3/28.

Question 43Data analysis, probability and statistics

The mean of a data set is 25 and the median is 25. Which feature of the distribution is suggested by this information?

  1. The distribution is roughly symmetric Correct answer.
  2. The distribution is strongly skewed
  3. The distribution has zero range
  4. The distribution has no mode

Explanation

When the mean and median are approximately equal, the distribution is generally close to symmetric because neither tail dominates. Skewness would typically separate the mean from the median.

Question 44Data analysis, probability and statistics

A probability distribution for a random variable X is given: P(X = 0) = 0.1, P(X = 1) = 0.4, P(X = 2) = 0.3 and P(X = 3) = p. What is the expected value of X?

  1. 2.0
  2. 1.7 Correct answer.
  3. 1.5
  4. 1.2

Explanation

The probabilities must sum to 1, so p equals 0.2. The expected value is 0 times 0.1 plus 1 times 0.4 plus 2 times 0.3 plus 3 times 0.2, which totals 1.7.

Question 45Data analysis, probability and statistics

A set of 9 values is arranged in order and the median is 40. The lower quartile is 25 and the upper quartile is 55. A value of 5 is recorded. Using the 1.5 times interquartile range rule, which statement is correct?

  1. 5 is an outlier because it is below the lower quartile
  2. 5 is an outlier because it is below the lower fence of -20
  3. 5 is not an outlier because it is above the lower fence Correct answer.
  4. 5 is not an outlier because the range is large

Explanation

The interquartile range is 55 minus 25, which is 30, and 1.5 times this is 45. The lower fence is 25 minus 45, which is -20, and 5 lies above -20, so it is not an outlier by this rule.

Question 46Data analysis, probability and statistics

Two independent events A and B have P(A) = 0.6 and P(B) = 0.5. What is the probability that exactly one of the two events occurs?

  1. 0.5 Correct answer.
  2. 0.2
  3. 0.8
  4. 0.3

Explanation

Exactly one occurs when A happens and B does not, or B happens and A does not. That is 0.6 times 0.5 plus 0.5 times 0.4, giving 0.3 plus 0.2, which equals 0.5.

Question 47Data analysis, probability and statistics

A data set is transformed by multiplying every value by 3. How do the mean and the standard deviation change?

  1. Both are increased by 3
  2. Both are multiplied by 3 Correct answer.
  3. The mean is unchanged and the standard deviation is multiplied by 3
  4. The mean is multiplied by 3 and the standard deviation is unchanged

Explanation

Multiplying every value by a constant multiplies both the centre and the spread by that constant. So the mean and the standard deviation are each multiplied by 3.

Question 48Data analysis, probability and statistics

A bag contains 6 red and 4 blue balls. Two balls are drawn without replacement. Given that the second ball drawn is blue, what is the probability that the first ball drawn was red?

  1. 2/3 Correct answer.
  2. 1/3
  3. 4/9
  4. 3/5

Explanation

Using conditional probability, P(first red and second blue) is 6/10 times 4/9, which is 24/90. The probability the second is blue is 4/10, so the required probability is 24/90 divided by 4/10, giving 2/3.

Question 49Data analysis, probability and statistics

A data set has mean 50 and standard deviation 10. Every value is increased by 20 and then multiplied by 2. What are the new mean and standard deviation?

  1. Mean 140, standard deviation 20 Correct answer.
  2. Mean 140, standard deviation 10
  3. Mean 120, standard deviation 20
  4. Mean 120, standard deviation 10

Explanation

Adding 20 raises the mean to 70, and multiplying by 2 gives 140. The standard deviation is unaffected by adding 20, but multiplying by 2 doubles it to 20.

Question 50Data analysis, probability and statistics

A test has 10 questions, each with probability 0.5 of being answered correctly, independently. What is the probability of exactly 8 correct answers?

  1. 45/1024 Correct answer.
  2. 1/2
  3. 8/10
  4. 10/1024

Explanation

The number of ways to choose 8 correct answers from 10 is 45. Each specific sequence of 10 answers has probability 1/1024, so the probability is 45/1024.