Statistics and Probability practice questions.

A hypothesis test at the 5% significance level gives a p-value of 0.03. The correct conclusion is to

• reject H₀, since 0.03 < 0.05
• accept H₀ as proven true
• reject H₀, since 0.03 < 0.5
• fail to reject H₀, since 0.03 < 0.05

Answer: reject H₀, since 0.03 < 0.05

Reject H₀ when p-value < significance level: 0.03 < 0.05, so reject. Note we never 'prove' a hypothesis — we either reject H₀ or fail to reject it, and always compare p with the stated significance level.

For a bivariate data set, the Pearson product-moment correlation coefficient is r = -0.92. This indicates

• a strong positive linear relationship
• a weak negative linear relationship
• no relationship between the variables
• a strong negative linear relationship

Answer: a strong negative linear relationship

|r| close to 1 means strong LINEAR association; the negative sign means y tends to decrease as x increases. Always mention both strength and direction, and the word 'linear'.

Calls arrive at a helpdesk at an average rate of 4 per hour, following a Poisson distribution. The probability of exactly 2 calls in an hour is closest to

• 0.147
• 0.183
• 0.195
• 0.238

Answer: 0.147

P(X = 2) = e⁻⁴ × 4²/2! = 8e⁻⁴ ≈ 0.147. Remember the formula P(X = k) = e^(−m) m^k / k!; 0.195 is P(X = 3) and 0.0183 is e⁻⁴ alone.

A two-sample t-test compares the mean yields of two fertilisers. The null hypothesis H₀ states that

• the two sample means are equal
• the two population means are equal
• the two population variances are equal
• the correlation between the samples is zero

Answer: the two population means are equal

Hypotheses are always statements about POPULATION parameters: H₀: μ₁ = μ₂. The sample means will almost never be exactly equal — the test asks whether their difference is significant.

A school divides its students into year groups and then selects a random sample from each year group, proportional to the size of the group. This sampling method is

• simple random sampling
• systematic sampling
• stratified sampling
• convenience sampling

Answer: stratified sampling

Splitting the population into non-overlapping subgroups (strata) and sampling randomly within each, in proportion, is stratified sampling. Systematic sampling would take every kth student from a list.

Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.4. Find P(A ∪ B).

• 0.12
• 0.58
• 0.7
• 1

Answer: 0.7

Mutually exclusive means P(A ∩ B) = 0, so P(A ∪ B) = 0.3 + 0.4 = 0.7. (0.58 would be the answer if A and B were independent instead.)