Statistics & Probability

Find the mean of the data set 4, 8, 10, 14, 14.

• A12
• B9
• C14
• D10✓

Answer: The mean is (4 + 8 + 10 + 14 + 14)/5 = 50/5 = 10.

A fair six-sided die is rolled once. Find the probability of obtaining a prime number.

• A1/6
• B2/3
• C1/2✓
• D1/3

Answer: The primes on a die are 2, 3 and 5, giving 3 favourable outcomes out of 6, so the probability is 3/6 = 1/2.

The random variable X follows a binomial distribution B(10, 0.3). Find E(X), the expected value.

• A7
• B3✓
• C3.3
• D0.3

Answer: For X ~ B(n, p), E(X) = np = 10 × 0.3 = 3.

The heights of a population are normally distributed with mean 170 cm. What proportion of the population is taller than 170 cm?

• A0.34
• B1
• C0.68
• D0.5✓

Answer: The normal distribution is symmetric about its mean, so exactly half (0.5) of the population lies above the mean of 170 cm.

A school has 1200 students split into Year 12 (700 students) and Year 13 (500 students). A researcher wants a sample of 60 students that reflects this split, taking 35 from Year 12 and 25 from Year 13, choosing randomly within each year group. Which sampling method is this?

• ASystematic sampling
• BStratified sampling✓
• CQuota sampling
• DOpportunity sampling

Answer: The population is divided into groups (strata) and a number is taken from each in proportion to its size (700:500 = 35:25), with random selection within each group. That is stratified sampling. Quota sampling would not use random selection within the groups.

Find the population variance of the data set 5, 7, 9, 11, 13.

• A8✓
• B10
• C2√2
• D40

Answer: The mean is (5+7+9+11+13)/5 = 9. The squared deviations are 16, 4, 0, 4, 16, summing to 40. The variance is 40/5 = 8. The value 2√2 is √8, the standard deviation rather than the variance; 40 is the sum of squared deviations before dividing by n; and 10 comes from dividing by n − 1 = 4, the sample formula rather than the population one.

The ordered data set is 4, 6, 7, 9, 11, 12, 14, 15, 18, 21, 25 (11 values). Using the positions (n+1)/4 and 3(n+1)/4 for the quartiles, find the interquartile range.

• A9
• B12
• C6
• D11✓

Answer: With n = 11, the lower quartile is the 3rd value Q1 = 7 and the upper quartile is the 9th value Q3 = 18. The interquartile range is Q3 − Q1 = 18 − 7 = 11.

For a data set the lower quartile is 15 and the upper quartile is 27. Using the 1.5 × IQR rule, which of the following values would be classed as an outlier?

• A44
• B10
• C46✓
• D5

Answer: The IQR is 27 − 15 = 12, so 1.5 × IQR = 18. The fences are Q1 − 18 = −3 (lower) and Q3 + 18 = 45 (upper). A value is an outlier if it is below −3 or above 45. Only 46 lies outside this range; 44, 10 and 5 all lie within the fences.

On any day the probability it rains is 0.3. If it rains, the probability a train is late is 0.6; if it does not rain, the probability the train is late is 0.1. Given that the train was late, find the probability that it had rained.

• A0.18
• B0.60
• C0.25
• D0.72✓

Answer: P(late) = P(rain)·P(late|rain) + P(no rain)·P(late|no rain) = 0.3×0.6 + 0.7×0.1 = 0.18 + 0.07 = 0.25. By conditional probability, P(rain|late) = P(rain and late)/P(late) = 0.18/0.25 = 0.72.

In a group of 30 students, 18 study French, 15 study German and 7 study both subjects. How many students study neither French nor German?

• A4✓
• B0
• C11
• D7

Answer: By the addition rule, the number studying at least one subject is 18 + 15 − 7 = 26. So the number studying neither is 30 − 26 = 4. (Subtracting the overlap of 7 once avoids double-counting.)

A scatter diagram of two variables gives a product-moment correlation coefficient of r = −0.92. Which statement best interprets this value?

• AThere is a strong positive linear correlation
• BThere is a strong negative linear correlation✓
• CThere is no linear correlation
• DOne variable causes the other to decrease

Answer: The PMCC lies between −1 and +1. A value of −0.92 is close to −1, indicating a strong negative linear correlation: as one variable increases the other tends to decrease. Correlation does not by itself establish causation, so concluding that one variable causes the other to decrease is not justified.

Two events A and B are independent with P(A) = 0.6 and P(B) = 0.3. Find P(A ∪ B).

• A0.90
• B0.18
• C0.72✓
• D0.78

Answer: For independent events P(A ∩ B) = P(A)·P(B) = 0.6 × 0.3 = 0.18. Then P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.3 − 0.18 = 0.72. Adding the probabilities without subtracting the intersection gives 0.90, while 0.18 is the intersection on its own.

Find the mean of the data set 2, 4, 4, 6, 9.

• A25
• B4
• C7
• D5✓

Answer: Add the values and divide by how many there are: 25 ÷ 5 = 5. The median and the range are offered too — all three are 'averages or spread', so read which one is asked for.

Find the mean of the data set 10, 12, 14, 16, 23.

• A75
• B14
• C13
• D15✓

Answer: Add the values and divide by how many there are: 75 ÷ 5 = 15. The median and the range are offered too — all three are 'averages or spread', so read which one is asked for.

Find the mean of the data set 1, 3, 5, 7, 14.

• A6✓
• B5
• C13
• D30

Answer: Add the values and divide by how many there are: 30 ÷ 5 = 6. The median and the range are offered too — all three are 'averages or spread', so read which one is asked for.

A and B are independent events with P(A) = 1/2 and P(B) = 1/3. Find P(A ∩ B).

• A2/3
• B5/6
• C1/6✓
• D3/2

Answer: For independent events P(A ∩ B) = P(A) × P(B) = 1/2 × 1/3 = 1/6. P(A ∪ B), the union, is offered too — 'and' multiplies, 'or' adds and then subtracts the overlap.

A and B are independent events with P(A) = 2/5 and P(B) = 1/4. Find P(A ∩ B).

• A13/20
• B1/10✓
• C11/20
• D8/5

Answer: For independent events P(A ∩ B) = P(A) × P(B) = 2/5 × 1/4 = 1/10. P(A ∪ B), the union, is offered too — 'and' multiplies, 'or' adds and then subtracts the overlap.

Find the variance of 2, 5, 6, 9, 13 (treating it as the whole population).

• A7
• B√(14)
• C14✓
• D11

Answer: Mean = 7. Variance is the mean of the squared deviations: Σ(x − x̄)²/n = 14. The standard deviation is offered too — it is the square root of the variance, not the variance itself.

Find the variance of 3, 7, 7, 11, 12 (treating it as the whole population).

• A2√(65)/5
• B52/5✓
• C8
• D9

Answer: Mean = 8. Variance is the mean of the squared deviations: Σ(x − x̄)²/n = 52/5. The standard deviation is offered too — it is the square root of the variance, not the variance itself.

A bag holds 4 red and 6 blue counters. Two are drawn without replacement. Find P(both red).

• A1/5
• B4/25
• C2/15✓
• D8/45

Answer: Without replacement the second probability changes: (4/10) × (3/9) = 12/90 = 2/15. (4/10)² is offered too — that would be the answer with replacement, where the bag is restored between draws.

What does a product moment correlation coefficient of −0.92 indicate?

• AA weak negative correlation
• BA strong negative linear correlation✓
• CNo correlation
• DA strong positive correlation once the sign is ignored

Answer: The magnitude gives the strength and the sign gives the direction: 0.92 is close to 1, so the linear relationship is strong, and the minus sign means one variable falls as the other rises. Correlation is not causation, and r only measures LINEAR association.

What is the range of a data set?

• AThe average of all the values added together and divided by the count
• BThe value that sits exactly in the middle once the data has been ordered
• CThe value which occurs most frequently throughout the whole data set
• DLargest value minus smallest value✓

Answer: Range = max − min, the simplest measure of spread. The mean, the median and the mode are all offered too — those are measures of location rather than spread.