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DSE M1 Sampling and Estimation Practice

DSE M1 · 60 practice questions with instant marking and worked solutions.

Sampling and Estimation is an important part of the DSE M1 mathematics curriculum. This topic develops the distribution of the sample mean, unbiased estimates, confidence intervals and the sample size needed. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Sampling and Estimation has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Sampling and Estimation; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Extended Part style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Sampling and Estimation set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Sampling and Estimation is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    A population has standard deviation 24; a random sample of size 9 is taken. Find the standard deviation of the sample mean. Choose the correct answer.
  2. 2 Basic
    A random sample of size 36 from a normal population with standard deviation 12 has mean 50. Find the lower limit of the 95% confidence interval for the population mean. Give the answer only.

  3. 3 Basic
    A random sample from a population is 6, 9, 18, 11, 16. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  4. 4 Basic
    A random sample of size $n$ is taken from a normal population with standard deviation 10; the 95% confidence interval for the population mean has width 3.92. The value of $n$ is 10. Is this true?
  5. 5 Basic
    A population has standard deviation 30; a random sample of size 25 is taken. Find the standard deviation of the sample mean. Choose the correct answer.
  6. 6 Basic
    A random sample of size 16 from a normal population with standard deviation 8 has mean 168. Find the upper limit of the 95% confidence interval for the population mean. Give the answer only.

  7. 7 Basic
    A random sample from a population is 16, 6, 20, 14, 16. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  8. 8 Basic
    A random sample of size $n$ is taken from a normal population with standard deviation 10; the 95% confidence interval for the population mean has width 7.84. The value of $n$ is 5. Is this true?
  9. 9 Basic
    The sample observations are independent and identically distributed. A population has standard deviation 15; a random sample of size 25 is taken. Find the standard deviation of the sample mean. Choose the correct answer.
  10. 10 Basic
    A random sample of size 16 from a normal population with standard deviation 8 has mean 50. Find the lower limit of the 95% confidence interval for the population mean. Give the answer only.

  11. 11 Basic
    The sample observations are independent and identically distributed. A random sample from a population is 15, 6, 18, 2, 18. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  12. 12 Basic
    A random sample of size $n$ is taken from a normal population with standard deviation 4; the 95% confidence interval for the population mean has width 3.136. The value of $n$ is 5. Is this true?
  13. 13 Standard
    A population has standard deviation 24; a random sample of size 25 is taken. Find the standard deviation of the sample mean (as a fraction in lowest terms). Choose the correct answer.
  14. 14 Standard
    A random sample of size 100 from a normal population with standard deviation 20 has mean 168. Find the upper limit of the 95% confidence interval for the population mean. Give the answer only.

  15. 15 Standard
    A random sample from a population is 17, 17, 16, 12, 17. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  16. 16 Standard DSE long question
    Sample size 36, sample mean 150, population standard deviation 12. Answer each part; give the answer only.
    1. (a) Find the standard deviation of the sample mean.
    2. (b) Find the lower limit of the 95% confidence interval.
    3. (c) Find the upper limit of the 95% confidence interval.
  17. 17 Standard
    A population has standard deviation 15; a random sample of size 4 is taken. Find the standard deviation of the sample mean (as a fraction in lowest terms). Choose the correct answer.
  18. 18 Standard
    A random sample of size 16 from a normal population with standard deviation 8 has mean 62. Find the lower limit of the 95% confidence interval for the population mean. Give the answer only.

  19. 19 Standard
    A random sample from a population is 15, 13, 13, 15, 3. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  20. 20 Standard DSE long question
    Sample size 25, sample mean 50, population standard deviation 10. Answer each part; give the answer only.
    1. (a) Find the standard deviation of the sample mean.
    2. (b) Find the lower limit of the 95% confidence interval.
    3. (c) Find the upper limit of the 95% confidence interval.
  21. 21 Standard
    The sample observations are independent and identically distributed. A population has standard deviation 8; a random sample of size 16 is taken. Find the standard deviation of the sample mean. Choose the correct answer.
  22. 22 Standard
    A random sample of size 25 from a normal population with standard deviation 15 has mean 50. Find the upper limit of the 95% confidence interval for the population mean. Give the answer only.

  23. 23 Standard
    The sample observations are independent and identically distributed. A random sample from a population is 2, 6, 14, 7, 20. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  24. 24 Standard DSE long question
    An independent random sample is taken from a normal population. Sample size 16, sample mean 72, population standard deviation 8. Answer each part; give the answer only.
    1. (a) Find the standard deviation of the sample mean.
    2. (b) Find the lower limit of the 95% confidence interval.
    3. (c) Find the upper limit of the 95% confidence interval.
  25. 25 Advanced
    A random sample of size $n$ is taken from a normal population with standard deviation 6; the 95% confidence interval for the population mean has width 5.88. Find the value of $n$. Choose the correct answer.
  26. 26 Advanced
    A population has standard deviation 20; a random sample of size 36 is taken. Find the standard deviation of the sample mean (as a fraction in lowest terms). Give the answer only.

  27. 27 Advanced
    A random sample of size 36 from a normal population with standard deviation 12 has mean 62. The lower limit of the 95% confidence interval for the population mean is 58.08. Is this true?
  28. 28 Advanced DSE long question
    $X\sim N(60,15^2)$ and $P(Z<1)=0.8413$. Answer each part; give the answer only.
    1. (a) Find the standard score of $x=75$.
    2. (b) Find $P(X>75)$.
    3. (c) Find $P(45
  29. 29 Advanced
    A random sample of size $n$ is taken from a normal population with standard deviation 20; the 95% confidence interval for the population mean has width 9.8. Find the value of $n$. Choose the correct answer.
  30. 30 Advanced
    A population has standard deviation 8; a random sample of size 9 is taken. Find the standard deviation of the sample mean (as a fraction in lowest terms). Give the answer only.

  31. 31 Advanced
    A random sample of size 9 from a normal population with standard deviation 6 has mean 120. The upper limit of the 95% confidence interval for the population mean is 123.92. Is this true?
  32. 32 Advanced DSE long question
    $X\sim N(60,5^2)$ and $P(Z<2)=0.9772$. Answer each part; give the answer only.
    1. (a) Find the standard score of $x=70$.
    2. (b) Find $P(X>70)$.
    3. (c) Find $P(50
  33. 33 Advanced
    A random sample of size $n$ is taken from a normal population with standard deviation 12; the 95% confidence interval for the population mean has width 11.76. Find the value of $n$. Choose the correct answer.
  34. 34 Advanced
    The sample observations are independent and identically distributed. A population has standard deviation 20; a random sample of size 4 is taken. Find the standard deviation of the sample mean. Give the answer only.

  35. 35 Advanced
    A random sample of size 100 from a normal population with standard deviation 20 has mean 62. The upper limit of the 95% confidence interval for the population mean is 65.92. Is this true?
  36. 36 Advanced DSE long question
    $X\sim N(70,5^2)$ and $P(Z<2)=0.9772$. Answer each part; give the answer only.
    1. (a) Find the standard score of $x=80$.
    2. (b) Find $P(X>80)$.
    3. (c) Find $P(60
  37. 37 Challenging
    A random sample from a population is 9, 15, 18, 8, 16. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Choose the correct answer.
  38. 38 Challenging
    A random sample of size $n$ is taken from a normal population with standard deviation 15; the 95% confidence interval for the population mean has width 5.88. Find the value of $n$. Show the essential working and give the answer.

  39. 39 Challenging
    A population has standard deviation 30; a random sample of size 16 is taken. The standard deviation of the sample mean is 15/8. Is this true?
  40. 40 Challenging DSE long question
    Sample size 16, sample mean 50, population standard deviation 8. Answer each part; give the answer only.
    1. (a) Find the standard deviation of the sample mean.
    2. (b) Find the lower limit of the 95% confidence interval.
    3. (c) Find the upper limit of the 95% confidence interval.
  41. 41 Challenging
    A random sample from a population is 19, 19, 17, 10, 17. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Choose the correct answer.
  42. 42 Challenging
    A random sample of size $n$ is taken from a normal population with standard deviation 5; the 95% confidence interval for the population mean has width 4.9. Find the value of $n$. Show the essential working and give the answer.

  43. 43 Challenging
    A population has standard deviation 24; a random sample of size 4 is taken. The standard deviation of the sample mean is 6. Is this true?
  44. 44 Challenging DSE long question
    Sample size 25, sample mean 150, population standard deviation 10. Answer each part; give the answer only.
    1. (a) Find the standard deviation of the sample mean.
    2. (b) Find the lower limit of the 95% confidence interval.
    3. (c) Find the upper limit of the 95% confidence interval.
  45. 45 Challenging
    The sample observations are independent and identically distributed. A random sample from a population is 13, 6, 12, 5, 2. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Choose the correct answer.
  46. 46 Challenging
    A random sample of size $n$ is taken from a normal population with standard deviation 10; the 95% confidence interval for the population mean has width 7.84. Find the value of $n$. Show the essential working and give the answer.

  47. 47 Challenging
    The sample observations are independent and identically distributed. A population has standard deviation 30; a random sample of size 25 is taken. The standard deviation of the sample mean is 6/5. Is this true?
  48. 48 Challenging DSE long question
    An independent random sample is taken from a normal population. Sample size 36, sample mean 50, population standard deviation 12. Answer each part; give the answer only.
    1. (a) Find the standard deviation of the sample mean.
    2. (b) Find the lower limit of the 95% confidence interval.
    3. (c) Find the upper limit of the 95% confidence interval.
  49. 49 Elite
    A random sample of size 25 from a normal population with standard deviation 10 has mean 120. Find the upper limit of the 95% confidence interval for the population mean. Give the answer only.

  50. 50 Elite
    A random sample from a population is 8, 15, 6, 16, 11. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  51. 51 Elite
    A random sample of size $n$ is taken from a normal population with standard deviation 20; the 95% confidence interval for the population mean has width 9.8. The value of $n$ is 64. Is this true?
  52. 52 Elite DSE long question
    $X\sim N(100,15^2)$ and $P(Z<2)=0.9772$. Answer each part; give the answer only.
    1. (a) Find the standard score of $x=130$.
    2. (b) Find $P(X>130)$.
    3. (c) Find $P(70
  53. 53 Elite
    A random sample of size 100 from a normal population with standard deviation 20 has mean 75. Find the lower limit of the 95% confidence interval for the population mean. Give the answer only.

  54. 54 Elite
    A random sample from a population is 11, 17, 18, 7, 16. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  55. 55 Elite
    A random sample of size $n$ is taken from a normal population with standard deviation 15; the 95% confidence interval for the population mean has width 5.88. The value of $n$ is 100. Is this true?
  56. 56 Elite DSE long question
    $X\sim N(60,15^2)$ and $P(Z<2)=0.9772$. Answer each part; give the answer only.
    1. (a) Find the standard score of $x=90$.
    2. (b) Find $P(X>90)$.
    3. (c) Find $P(30
  57. 57 Elite
    A random sample of size 9 from a normal population with standard deviation 6 has mean 168. Find the upper limit of the 95% confidence interval for the population mean. Give the answer only.

  58. 58 Elite
    The sample observations are independent and identically distributed. A random sample from a population is 9, 5, 9, 3, 18. Find the unbiased estimate $s^2$ of the population variance (as a fraction in lowest terms). Show the essential working and give the answer.

  59. 59 Elite
    A random sample of size $n$ is taken from a normal population with standard deviation 5; the 95% confidence interval for the population mean has width 4.9. The value of $n$ is 16. Is this true?
  60. 60 Elite DSE long question
    $X\sim N(70,5^2)$ and $P(Z<1)=0.8413$. Answer each part; give the answer only.
    1. (a) Find the standard score of $x=75$.
    2. (b) Find $P(X>75)$.
    3. (c) Find $P(65

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Common mistakes

Common mistakes in Sampling and Estimation

This topic needs particular care with the distribution of the sample mean, unbiased estimates, confidence intervals and the sample size needed. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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