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← Mathematical Statistics
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Practice deliberately

Work through Mathematical Statistics unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Mathematical Statistics coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Mathematical Statistics syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use statistics practice questions with answers.

01

Sampling distributions

Define the experiment and population, select the correct probability model, calculate with its assumptions visible, and interpret uncertainty in context.

Goal: recognise a sampling distributions problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Point estimation

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a point estimation problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Interval estimation

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a interval estimation problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Hypothesis tests

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a hypothesis tests problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Likelihood

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a likelihood problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Regression

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a regression problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Bayesian foundations

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a bayesian foundations problem from its wording, carry out the governing method, and check that the result is reasonable.

Mathematical Statistics questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: For a sample of 100 with a true proportion of 0.4, what is the standard error of the sample proportion, to three decimal places?

Mathematical Statistics · Sampling distributions

  1. 0.049
  2. 0.240
  3. 0.004
  4. 0.400
Show the worked answer
  1. The standard error of a proportion is the square root of p(1 - p) / n.
  2. p(1 - p) = 0.4 x 0.6 = 0.24.
  3. 0.24 / 100 = 0.0024.
  4. The square root of 0.0024 is 0.049.

Answer: 0.049

Question 2: An estimator has variance 9 and bias 2. What is its mean squared error?

Mathematical Statistics · Point estimation

  1. 13
  2. 11
  3. 18
  4. 9
Show the worked answer
  1. Mean squared error equals variance plus the square of the bias.
  2. The bias squared is 2^2 = 4.
  3. 9 + 4 = 13.
  4. An unbiased estimator would have MSE equal to its variance, 9.

Answer: 13

Question 3: A sample of 100 has a mean of 120 and a known standard deviation of 15. What is the 99% confidence interval, to two decimal places?

Mathematical Statistics · Interval estimation

  1. (116.14, 123.86)
  2. (117.06, 122.94)
  3. (105.00, 135.00)
  4. (118.50, 121.50)
Show the worked answer
  1. The standard error is 15 divided by the square root of 100, which is 1.5.
  2. For 99% confidence, z = 2.576.
  3. The margin of error is 2.576 x 1.5 = 3.864.
  4. 120 - 3.864 = 116.14 and 120 + 3.864 = 123.86.

Answer: (116.14, 123.86)

Question 4: Test H0: mu = 50 against a two-sided alternative with sample mean 52, sigma = 8, and n = 64. What is the z statistic?

Mathematical Statistics · Hypothesis tests

  1. 2.0
  2. 0.25
  3. 16.0
  4. 1.5
Show the worked answer
  1. The z statistic is (sample mean minus the claimed mean) divided by the standard error.
  2. The standard error is 8 divided by the square root of 64, which is 8 / 8 = 1.
  3. The difference is 52 - 50 = 2.
  4. 2 / 1 = 2.0, which gives a two-sided p-value of about 0.046.

Answer: 2.0

Question 5: Twenty independent Bernoulli trials give 7 successes. What is the maximum likelihood estimate of p?

Mathematical Statistics · Likelihood

  1. 0.35
  2. 0.50
  3. 0.29
  4. 0.65
Show the worked answer
  1. The likelihood is p^7 times (1 - p)^13.
  2. Taking logs gives 7 ln p + 13 ln(1 - p).
  3. Setting the derivative to zero gives 7 / p = 13 / (1 - p), so 7 - 7p = 13p and 20p = 7.
  4. p = 7 / 20 = 0.35, which is just the sample proportion.

Answer: 0.35

Question 6: A regression has Sxy = 84, Sxx = 28, mean x = 4, and mean y = 20. What are the slope and intercept?

Mathematical Statistics · Regression

  1. slope 3, intercept 8
  2. slope 3, intercept 20
  3. slope 0.33, intercept 18.7
  4. slope 2, intercept 12
Show the worked answer
  1. The least-squares slope is Sxy divided by Sxx.
  2. 84 / 28 = 3.
  3. The line passes through the point of means, so intercept = mean y minus slope times mean x.
  4. 20 - 3(4) = 20 - 12 = 8.

Answer: slope 3, intercept 8

Question 7: A Beta(2, 3) prior is updated with 6 successes and 4 failures. What is the posterior mean, to four decimal places?

Mathematical Statistics · Bayesian foundations

  1. 0.5333
  2. 0.6000
  3. 0.4000
  4. 0.5000
Show the worked answer
  1. The Beta prior is conjugate: add successes to the first parameter and failures to the second.
  2. The posterior is Beta(2 + 6, 3 + 4) = Beta(8, 7).
  3. The mean of Beta(a, b) is a / (a + b).
  4. 8 / 15 = 0.5333.

Answer: 0.5333

Question 8: A sample of 16 has standard deviation 4. What is the estimated standard error of the mean, and how many degrees of freedom does the t procedure use?

Mathematical Statistics · Sampling distributions

  1. standard error 1, 15 degrees of freedom
  2. standard error 1, 16 degrees of freedom
  3. standard error 4, 15 degrees of freedom
  4. standard error 0.25, 15 degrees of freedom
Show the worked answer
  1. The estimated standard error is s divided by the square root of n.
  2. The square root of 16 is 4, so 4 / 4 = 1.
  3. A one-sample t procedure uses n - 1 degrees of freedom.
  4. 16 - 1 = 15.

Answer: standard error 1, 15 degrees of freedom

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Mathematical Statistics questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Sampling distributionsFor a sample of 100 with a true proportion of 0.4, what is the standard error of the sample proportion, to three decimal places?0.049
2Point estimationAn estimator has variance 9 and bias 2. What is its mean squared error?13
3Interval estimationA sample of 100 has a mean of 120 and a known standard deviation of 15. What is the 99% confidence interval, to two decimal places?(116.14, 123.86)
4Hypothesis testsTest H0: mu = 50 against a two-sided alternative with sample mean 52, sigma = 8, and n = 64. What is the z statistic?2.0
5LikelihoodTwenty independent Bernoulli trials give 7 successes. What is the maximum likelihood estimate of p?0.35
6RegressionA regression has Sxy = 84, Sxx = 28, mean x = 4, and mean y = 20. What are the slope and intercept?slope 3, intercept 8
7Bayesian foundationsA Beta(2, 3) prior is updated with 6 successes and 4 failures. What is the posterior mean, to four decimal places?0.5333
8Sampling distributionsA sample of 16 has standard deviation 4. What is the estimated standard error of the mean, and how many degrees of freedom does the t procedure use?standard error 1, 15 degrees of freedom

How the generated Mathematical Statistics sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Mathematical Statistics practice page

What does this page cover?

It covers all 7 Mathematical Statistics units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.