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BCS Written Statistics study materials
BCS Written Statistics study resources

BCS Written Statistics practice questions

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Question 1Probability Theory and Probability Distributions

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

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

Explanation

The outcomes greater than 4 are 5 and 6, giving two favourable outcomes out of six equally likely outcomes. The probability is therefore 2/6, which simplifies to 1/3.

Question 2Probability Theory and Probability Distributions

For any two events A and B, which expression gives the addition rule for P(A ∪ B)?

  1. P(A) + P(B) − P(A ∩ B) Correct answer.
  2. P(A) × P(B)
  3. P(A) + P(B) + P(A ∩ B)
  4. P(A) + P(B)

Explanation

The union probability equals the sum of the individual probabilities minus the probability of the overlap, because the intersection is counted twice when the two probabilities are added. Subtracting P(A ∩ B) once corrects this double counting.

Question 3Probability Theory and Probability Distributions

Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.5. What is P(A ∪ B)?

  1. 0.8 Correct answer.
  2. 0.15
  3. 1.0
  4. 0.2

Explanation

Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. The addition rule then reduces to P(A) + P(B) = 0.3 + 0.5 = 0.8.

Question 4Probability Theory and Probability Distributions

Two events A and B are independent. Which condition must hold?

  1. P(A ∩ B) = 0
  2. P(A ∩ B) = P(A)P(B) Correct answer.
  3. P(A | B) = P(B)
  4. P(A ∪ B) = P(A) + P(B)

Explanation

Independence means the occurrence of one event does not change the probability of the other, which is equivalent to the multiplication rule P(A ∩ B) = P(A)P(B). This definition also implies P(A | B) = P(A) whenever P(B) > 0.

Question 5Probability Theory and Probability Distributions

A bag contains 4 red and 6 blue balls. One ball is drawn at random. What is the probability that it is blue?

  1. 0.24
  2. 0.5
  3. 0.4
  4. 0.6 Correct answer.

Explanation

There are 10 balls in total and 6 of them are blue, so the probability is 6/10 = 0.6. Each ball is equally likely to be selected.

Question 6Probability Theory and Probability Distributions

In a discrete probability distribution, what must the sum of all probabilities over the sample space equal?

  1. 1 Correct answer.
  2. 0.5
  3. The number of outcomes
  4. 0

Explanation

A valid probability distribution assigns non-negative probabilities to each outcome and the total over the entire sample space must be exactly one. This normalisation ensures the full set of outcomes is certain to occur.

Question 7Probability Theory and Probability Distributions

X is a discrete random variable with P(X = 0) = 0.2, P(X = 1) = 0.5 and P(X = 2) = 0.3. What is E(X)?

  1. 1.5
  2. 0.9
  3. 1.0
  4. 1.1 Correct answer.

Explanation

The expectation is the sum of each value multiplied by its probability: 0(0.2) + 1(0.5) + 2(0.3) = 0 + 0.5 + 0.6 = 1.1.

Question 8Probability Theory and Probability Distributions

Which statement correctly describes the variance of a random variable?

  1. It is the expected squared deviation from the mean Correct answer.
  2. It measures the largest possible value
  3. It is the square root of the mean
  4. It is always equal to the mean

Explanation

Variance is defined as E[(X − μ)²], the average squared distance of the variable from its mean. It quantifies dispersion, and its square root is the standard deviation.

Question 9Probability Theory and Probability Distributions

A fair coin is tossed 10 times. What is the probability of exactly 5 heads?

  1. 10/1024
  2. 252/1024 Correct answer.
  3. 1/2
  4. 5/10

Explanation

The number of heads follows a binomial distribution with n = 10 and p = 0.5. The probability is C(10,5)(0.5)^10 = 252/1024, since C(10,5) = 252 and there are 2^10 equally likely sequences.

Question 10Probability Theory and Probability Distributions

A Poisson random variable has mean 3. What is P(X = 0)?

  1. 1 − e^-3
  2. 3e^-3
  3. e^-3 Correct answer.
  4. e^-1/3

Explanation

The Poisson probability mass function is P(X = k) = e^-λ λ^k / k!. Setting k = 0 and λ = 3 gives e^-3 × 3^0 / 0! = e^-3.

Question 11Probability Theory and Probability Distributions

If X is binomial with n trials and success probability p, what are E(X) and Var(X)?

  1. np and np(1 − p) Correct answer.
  2. n/p and n(1 − p)/p²
  3. np and np
  4. p and np(1 − p)

Explanation

A binomial variable is the sum of n independent Bernoulli trials each with mean p and variance p(1 − p). Hence the mean is np and the variance is np(1 − p).

Question 12Probability Theory and Probability Distributions

X is uniformly distributed on the interval [2, 8]. What is P(X < 5)?

  1. 0.2
  2. 0.375
  3. 0.5 Correct answer.
  4. 0.625

Explanation

The density is constant at 1/(8 − 2) = 1/6 over the interval. The probability below 5 is the length 5 − 2 = 3 times 1/6, giving 3/6 = 0.5.

Question 13Probability Theory and Probability Distributions

Z has a standard normal distribution. What is P(Z < 0)?

  1. 0.25
  2. 0
  3. 1
  4. 0.5 Correct answer.

Explanation

The standard normal density is symmetric about its mean of zero, so exactly half the probability lies below the mean. Hence P(Z < 0) = 0.5.

Question 14Probability Theory and Probability Distributions

A continuous random variable has density f(x) = 2x for 0 ≤ x ≤ 1 and zero elsewhere. What is E(X)?

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

Explanation

The expectation is the integral of x f(x) over the support: ∫₀¹ x(2x) dx = ∫₀¹ 2x² dx = 2/3. This is greater than 0.5 because the density increases with x.

Question 15Probability Theory and Probability Distributions

A screening test has sensitivity 0.9 and specificity 0.9, and the disease prevalence is 0.01. Approximately what is the probability that a person who tests positive actually has the disease?

  1. About 0.01
  2. About 0.50
  3. About 0.90
  4. About 0.08 Correct answer.

Explanation

Bayes' theorem gives P(disease | positive) = (0.9 × 0.01) / (0.9 × 0.01 + 0.1 × 0.99) = 0.009/0.108, which is roughly 0.08. The low prevalence makes false positives numerically dominant.

Question 16Probability Theory and Probability Distributions

For independent events A and B with P(A) = 0.4 and P(B) = 0.5, what is P(A ∪ B)?

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

Explanation

Independence gives P(A ∩ B) = 0.4 × 0.5 = 0.2. The addition rule then yields P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7.

Question 17Probability Theory and Probability Distributions

X has E(X) = 5 and Var(X) = 4. What is Var(3X + 2)?

  1. 38
  2. 36 Correct answer.
  3. 14
  4. 12

Explanation

For constants a and b, Var(aX + b) = a² Var(X), since adding a constant shifts the distribution without changing its spread. Here 3² × 4 = 36.

Question 18Probability Theory and Probability Distributions

A fair die is rolled repeatedly. What is the probability that the first 6 appears on the third roll?

  1. 1/216
  2. 1/6
  3. 25/216 Correct answer.
  4. 5/36

Explanation

The number of rolls until the first success follows a geometric distribution with p = 1/6. The probability of two failures followed by a success is (5/6)²(1/6) = 25/216.

Question 19Probability Theory and Probability Distributions

Which pair of parameters fully determines a normal distribution?

  1. Its variance and skewness
  2. Its median and mode
  3. Its mean and variance Correct answer.
  4. Its mean and range

Explanation

A normal distribution is specified by its location parameter μ and scale parameter σ². All other quantities, such as the mode, median and skewness, are then fixed by these two values.

Question 20Probability Theory and Probability Distributions

If X is normal with mean 10 and variance 4, what is P(X < 10)?

  1. 0.8413
  2. 0
  3. 0.5 Correct answer.
  4. 0.25

Explanation

The mean 10 is the centre of the distribution, and the normal curve is symmetric about its mean. Therefore half of the probability lies below 10, giving 0.5.

Question 21Probability Theory and Probability Distributions

X and Y are independent with E(X) = 2, E(Y) = 3, Var(X) = 1 and Var(Y) = 4. What is Var(X − Y)?

  1. 3
  2. 13
  3. 5 Correct answer.
  4. −3

Explanation

For independent variables, variances add even when the variables are subtracted: Var(X − Y) = Var(X) + Var(Y) = 1 + 4 = 5. The minus sign does not reduce the spread.

Question 22Probability Theory and Probability Distributions

A committee of 2 is chosen at random from 5 men and 3 women. What is the probability that both members are women?

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

Explanation

There are C(8,2) = 28 equally likely pairs in total and C(3,2) = 3 pairs consisting of two women. The probability is therefore 3/28.

Question 23Probability Theory and Probability Distributions

X has a binomial distribution with n = 50 and p = 0.04. Which approximation is most appropriate for computing P(X ≥ 2)?

  1. Binomial with n = 2
  2. Uniform on the integers 0 to 50
  3. Normal with mean 2 and variance 2
  4. Poisson with mean 2 Correct answer.

Explanation

When n is large and p is small, the binomial converges to a Poisson distribution with λ = np = 50 × 0.04 = 2. The Poisson approximation is accurate here because p is close to zero.

Question 24Probability Theory and Probability Distributions

A fair coin is tossed until the first head appears. What is the expected number of tosses?

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

Explanation

The number of tosses follows a geometric distribution with success probability p = 1/2, whose mean is 1/p. Hence the expected number of tosses is 2.

Question 25Probability Theory and Probability Distributions

X and Y are independent standard normal variables. What is the distribution of X + Y?

  1. Normal with mean 0 and variance 2 Correct answer.
  2. Standard normal
  3. Normal with mean 0 and variance 1
  4. Chi-squared with 2 degrees of freedom

Explanation

A sum of independent normal variables is itself normal, with means added and variances added. Since each variable has mean 0 and variance 1, X + Y is normal with mean 0 and variance 2.

Question 26Probability Theory and Probability Distributions

Which expression gives the conditional probability P(A | B) when P(B) > 0?

  1. P(A ∩ B) / P(A)
  2. P(A ∩ B) / P(B) Correct answer.
  3. P(A ∪ B) − P(B)
  4. P(A) P(B) / P(A ∪ B)

Explanation

Conditional probability rescales the joint probability of A and B by the probability of the conditioning event B, giving P(A | B) = P(A ∩ B)/P(B). This formula holds whenever P(B) > 0 and underlies the multiplication rule P(A ∩ B) = P(A | B)P(B).

Question 27Probability Theory and Probability Distributions

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

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

Explanation

The probability the first ball is red is 4/10 = 2/5. Given that, 3 red remain among 9 balls, so the second is red with probability 3/9 = 1/3. Multiplying gives (2/5)(1/3) = 2/15.

Question 28Probability Theory and Probability Distributions

For any two events A and B, which identity correctly expresses the addition rule?

  1. P(A ∪ B) = P(A) P(B) − P(A ∩ B)
  2. P(A ∪ B) = P(A) + P(B)
  3. P(A ∪ B) = P(A) + P(B) − P(A ∩ B) Correct answer.
  4. P(A ∪ B) = P(A) + P(B) + P(A ∩ B)

Explanation

Adding P(A) and P(B) counts the overlap A ∩ B twice, so the addition rule subtracts P(A ∩ B) once: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The simpler sum P(A) + P(B) is valid only when the events are mutually exclusive.

Question 29Probability Theory and Probability Distributions

A continuous random variable has cumulative distribution function F(x). Which expression equals P(a < X ≤ b) for a < b?

  1. F(b) / F(a)
  2. F(a) − F(b)
  3. F(b) + F(a)
  4. F(b) − F(a) Correct answer.

Explanation

By definition F(x) = P(X ≤ x), so the probability of falling between a and b is the difference of the cumulative values, F(b) − F(a). For continuous variables the endpoints carry zero probability, so P(a < X ≤ b) = P(a ≤ X ≤ b).

Question 30Probability Theory and Probability Distributions

X is a discrete random variable with P(X = −1) = 0.3, P(X = 0) = 0.2 and P(X = 2) = 0.5. What is E(X^2)?

  1. 0.7
  2. 1.9 Correct answer.
  3. 2.3
  4. 1.3

Explanation

The second moment is E(X^2) = Σ x^2 P(X = x) = (1)(0.3) + (0)(0.2) + (4)(0.5) = 0.3 + 0 + 2.0 = 2.3. Note that E(X) = −0.3 + 0 + 1.0 = 0.7, and Var(X) = 2.3 − 0.49 = 1.81.

Question 31Probability Theory and Probability Distributions

X is binomial with n = 4 and p = 0.5. What is P(X ≥ 3)?

  1. 1/4
  2. 3/8
  3. 5/16 Correct answer.
  4. 1/16

Explanation

P(X ≥ 3) = P(X = 3) + P(X = 4) = C(4,3)(0.5)^4 + C(4,4)(0.5)^4 = 4/16 + 1/16 = 5/16. Each term uses the binomial probability mass function with n = 4 and p = 0.5.

Question 32Probability Theory and Probability Distributions

X has a Poisson distribution with mean 2. What is P(X ≥ 1)?

  1. 1 − 2e^(−2)
  2. e^(−2)
  3. 1 − e^(−2) Correct answer.
  4. 2e^(−2)

Explanation

For a Poisson variable, P(X = 0) = e^(−λ) = e^(−2). Using the complement rule, P(X ≥ 1) = 1 − P(X = 0) = 1 − e^(−2), which is about 0.865.

Question 33Probability Theory and Probability Distributions

A continuous random variable has density f(x) = 3x^2 for 0 ≤ x ≤ 1 and zero elsewhere. What is P(X ≤ 0.5)?

  1. 0.25
  2. 0.125 Correct answer.
  3. 0.5
  4. 0.375

Explanation

Integrating the density from 0 to 0.5 gives ∫ 3x^2 dx = x^3 evaluated from 0 to 0.5, which equals 0.125. Equivalently, the cumulative distribution is F(x) = x^3 on the unit interval.

Question 34Probability Theory and Probability Distributions

X is normal with mean 5 and standard deviation 2. Which expression gives P(X > 7)?

  1. 1 − Φ(2)
  2. 1 − Φ(1) Correct answer.
  3. Φ(1)
  4. Φ(2)

Explanation

Standardize: z = (7 − 5)/2 = 1, so P(X > 7) = P(Z > 1) = 1 − Φ(1), about 0.159. The complement is needed because the event is a right tail.

Question 35Probability Theory and Probability Distributions

X and Y are independent random variables with E(X) = 1, E(Y) = 4, Var(X) = 3 and Var(Y) = 2. What is E(XY)?

  1. 11
  2. 4 Correct answer.
  3. 8
  4. 5

Explanation

Independence implies E(XY) = E(X)E(Y). Here that product is 1 × 4 = 4. The variances are irrelevant to the expectation of the product.

Question 36Probability Theory and Probability Distributions

X is geometric with success probability p on each independent trial. What is Var(X), counting the number of trials up to and including the first success?

  1. (1 − p)/p^2 Correct answer.
  2. 1/p
  3. (1 − p)/p
  4. p(1 − p)

Explanation

For this geometric variable, E(X) = 1/p and Var(X) = (1 − p)/p^2. The variance grows rapidly as p becomes small, reflecting the long waiting times that become likely.

Question 37Probability Theory and Probability Distributions

A test for a condition has sensitivity 0.95 and specificity 0.90, and the prevalence is 0.02. Given a positive result, what is the approximate probability the person has the condition?

  1. About 0.50
  2. About 0.02
  3. About 0.95
  4. About 0.16 Correct answer.

Explanation

By Bayes' theorem, P(D | +) = (0.95 × 0.02) / (0.95 × 0.02 + 0.10 × 0.98). The numerator is 0.019 and the denominator is 0.019 + 0.098 = 0.117, giving about 0.162. The low prevalence makes false positives dominate.

Question 38Probability Theory and Probability Distributions

X is uniform on [0, 1]. What is the variance of X?

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

Explanation

For a uniform variable on [a, b], the variance is (b − a)^2/12. With a = 0 and b = 1 this gives 1/12, about 0.083. The mean is 1/2 and E(X^2) = 1/3, so Var(X) = 1/3 − 1/4 = 1/12.

Question 39Probability Theory and Probability Distributions

X and Y are independent with Var(X) = 4 and Var(Y) = 9. What is Var(2X − 3Y)?

  1. 97 Correct answer.
  2. 85
  3. 13
  4. −65

Explanation

For independent variables, Var(aX + bY) = a^2 Var(X) + b^2 Var(Y). Here that is 4(4) + 9(9) = 16 + 81 = 97. Squaring the coefficients makes both contributions positive despite the minus sign.

Question 40Probability Theory and Probability Distributions

X and Y are independent exponential variables, each with mean 1. What is P(X > Y)?

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

Explanation

With identical independent exponential distributions, the joint density is symmetric in x and y. The events X > Y and Y > X are equally likely and the probability of a tie is zero, so each has probability 1/2.

Question 41Statistical Inference (Estimation and Hypothesis Testing)

A sample mean is used to estimate the population mean. Which property makes it an unbiased estimator?

  1. Its value is always within one standard error of the population mean
  2. Its sampling distribution is exactly normal for every sample size
  3. Its expected value over repeated sampling equals the population mean Correct answer.
  4. Its variance decreases as the population variance increases

Explanation

Unbiasedness is a property of the sampling distribution: the average of the estimates obtained across repeated samples equals the true parameter value. It says nothing about any single estimate being close to the truth, and it does not require normality of the sampling distribution.

Question 42Statistical Inference (Estimation and Hypothesis Testing)

A 95% confidence interval for a population mean is reported as (12.4, 18.6). Which interpretation is correct?

  1. The probability that the population mean lies in this specific interval is 0.95
  2. 95% of the population values lie between 12.4 and 18.6
  3. If the study were repeated many times, about 95% of intervals constructed this way would contain the true mean Correct answer.
  4. 95% of sample means lie between 12.4 and 18.6

Explanation

The confidence level describes the long-run performance of the procedure: about 95% of intervals built by this method capture the true parameter. Once a specific interval is computed, the parameter either lies inside it or does not, so no probability attaches to that single interval.

Question 43Statistical Inference (Estimation and Hypothesis Testing)

In a hypothesis test, the p-value is defined as which quantity?

  1. The probability that the null hypothesis is true given the data
  2. The probability that the alternative hypothesis is false
  3. The probability of obtaining a test statistic at least as extreme as the observed one, assuming the null hypothesis is true Correct answer.
  4. The probability of making a Type I error in this particular study

Explanation

The p-value is a tail probability computed under the assumption that the null hypothesis holds. It measures how unusual the observed statistic would be if the null were true, and it is not the probability that any hypothesis is true or false.

Question 44Statistical Inference (Estimation and Hypothesis Testing)

A test is conducted at the 5% significance level and the p-value is 0.03. What is the appropriate conclusion?

  1. Conclude that the effect size is large
  2. Increase the sample size before deciding
  3. Accept the null hypothesis as true
  4. Reject the null hypothesis at the 5% level Correct answer.

Explanation

The decision rule rejects the null hypothesis whenever the p-value is less than or equal to the chosen significance level. Since 0.03 is below 0.05, the null hypothesis is rejected in favour of the alternative.

Question 45Statistical Inference (Estimation and Hypothesis Testing)

Which statement correctly describes a Type II error?

  1. Failing to reject a false null hypothesis Correct answer.
  2. Rejecting a true null hypothesis
  3. Reporting a confidence interval that excludes the parameter
  4. Choosing the wrong significance level

Explanation

A Type II error occurs when the test fails to detect a real departure from the null hypothesis. Its probability is usually denoted beta, and its complement, one minus beta, is the power of the test.

Question 46Statistical Inference (Estimation and Hypothesis Testing)

Holding everything else constant, what happens to the width of a confidence interval for a mean when the sample size is increased?

  1. The interval becomes centred on zero
  2. The interval becomes wider
  3. The interval becomes narrower Correct answer.
  4. The interval width does not change

Explanation

The standard error of the mean equals the population standard deviation divided by the square root of the sample size, so it shrinks as the sample grows. A smaller standard error produces a narrower interval for the same confidence level.

Question 47Statistical Inference (Estimation and Hypothesis Testing)

The sampling distribution of the sample mean is approximately normal for large samples regardless of the population shape. Which theorem guarantees this?

  1. The law of iterated expectations
  2. The law of total probability
  3. The central limit theorem Correct answer.
  4. Chebyshev's inequality

Explanation

The central limit theorem states that sums and averages of independent, identically distributed observations tend to normality as the sample size grows, whatever the shape of the underlying population. This result underpins most large-sample inference for means.

Question 48Statistical Inference (Estimation and Hypothesis Testing)

Which quantity is the standard error of a sample proportion based on n independent observations, where p is the population proportion?

  1. sqrt(p(1 - p)) / n
  2. n p (1 - p)
  3. p(1 - p) / n
  4. sqrt(p(1 - p) / n) Correct answer.

Explanation

The sample proportion has variance p(1-p)/n, and the standard error is the square root of that variance. It decreases as the square root of the sample size, which governs the precision of the estimate.

Question 49Statistical Inference (Estimation and Hypothesis Testing)

A researcher wants to test whether a new teaching method changes average test scores, without assuming the direction of the change in advance. Which alternative hypothesis is appropriate?

  1. The population mean is less than the historical value
  2. The population mean differs from the historical value Correct answer.
  3. The population mean is greater than the historical value
  4. The population mean equals the historical value

Explanation

A two-sided alternative states only that the parameter differs from the null value, allowing departures in either direction. One-sided alternatives would require a directional claim established before seeing the data.

Question 50Statistical Inference (Estimation and Hypothesis Testing)

A 90% confidence interval for a mean is (40.2, 47.8). What is the correct interpretation of the confidence level?

  1. There is a 90% chance that the sample mean equals the population mean
  2. 90% of future sample means will fall in this interval
  3. The procedure that produced this interval captures the true mean in about 90% of repeated applications Correct answer.
  4. 90% of the population lies in this interval

Explanation

The confidence level is a property of the repeated-sampling procedure, describing the proportion of intervals that would contain the parameter if the study were repeated indefinitely. It does not describe probabilities for the fixed interval already observed.