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Sampling Distribution and Estimation · CAPE Applied Mathematics Unit 1

74 past-paper questions on Sampling Distribution and Estimation, part of Module 3: Analysing and Interpreting Data, from every CAPE Applied Mathematics Unit 1 paper on Quelpr.

Practise Sampling Distribution and Estimation
  1. 5(a)6 marks· CAPE Applied Mathematics Unit 1 · 2008 · Paper 2Omitting the continuity correction, calculate an approximate 97% confidence interval for p.
  2. 5(b)(ii)2 marks· CAPE Applied Mathematics Unit 1 · 2008 · Paper 2Justify the use of a normal distribution approximation in this large-sample test of a binomial proportion.
  3. 5(a)(i)a)2 marks· CAPE Applied Mathematics Unit 1 · 2008 (second paper) · Paper 2Calculate the unbiased estimate of the population mean.
  4. 5(a)(i)b)3 marks· CAPE Applied Mathematics Unit 1 · 2008 (second paper) · Paper 2Calculate the unbiased estimate of the population standard deviation.
  5. 5(a)2 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2Explain what is meant by the term "a 95% confidence interval" in the context of a population mean.
  6. 5(b)(i)2 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2Determine an unbiased estimate of the standard deviation.
  7. 5(b)(ii)4 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2Construct, without using a continuity correction, a 95% confidence interval for the mean amount of money saved.
  8. 5(b)(iii)a)2 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2If the consultant considers that a 95% confidence interval is too wide, state TWO methods that can be taken to reduce this width.
  9. 5(b)(iii)b)2 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2State which method is better and give a reason for your answer.
  10. 5(b)(iv)4 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2Determine the least number of farmers that must be sampled so that the width of a 99% confidence interval is less than 40.
  11. 5(c)2 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2A 90% confidence interval for population mean, μ, is found for each sample when 60 random samples of size 45 are taken. Determine the expected number of intervals that do NOT contain μ.
  12. 5(d)(i)2 marks· CAPE Applied Mathematics Unit 1 · 2009 (second paper) · Paper 2State the distribution modelled by X̄ giving its parameters.
  13. 5(a)2 marks· CAPE Applied Mathematics Unit 1 · 2010 · Paper 2State the distribution of X̄ giving its parameters.
  14. 5(b)(i)4 marks· CAPE Applied Mathematics Unit 1 · 2010 · Paper 2Calculate the value of X̄.
  15. 5(b)(ii)4 marks· CAPE Applied Mathematics Unit 1 · 2010 · Paper 2Calculate the number of observations taken, n.
  16. 5(c)2 marks· CAPE Applied Mathematics Unit 1 · 2010 · Paper 2If 20 independent samples, EACH of size n, are drawn from a distribution and the 90% confidence interval is calculated for EACH, how many of these intervals will NOT contain the population mean μ?
  17. 5(d)(i)a)3 marks· CAPE Applied Mathematics Unit 1 · 2010 · Paper 2Calculate an unbiased estimate of the population mean.
  18. 5(d)(i)b)5 marks· CAPE Applied Mathematics Unit 1 · 2010 · Paper 2Calculate an unbiased estimate of the population standard deviation.
  19. 5(d)(ii)5 marks· CAPE Applied Mathematics Unit 1 · 2010 · Paper 2Construct a 94% confidence interval for the mean amount of money spent by the students at the bookstore.
  20. 6(a)3 marks· CAPE Applied Mathematics Unit 1 · 2011 · Paper 2State the approximate distribution of X̄ giving its parameters.
  21. 5(a)(i)3 marks· CAPE Applied Mathematics Unit 1 · 2012 · Paper 2State the mean, the variance and the distribution of the mean, X̄, of the sample.
  22. 5(a)(ii)4 marks· CAPE Applied Mathematics Unit 1 · 2012 · Paper 2Calculate the probability that the sample mean is less than 417 grams.
  23. 5(b)4 marks· CAPE Applied Mathematics Unit 1 · 2012 · Paper 2Calculate a 98% confidence interval for the mean length of the pencils produced by the machine.
  24. 5(c)(i)4 marks· CAPE Applied Mathematics Unit 1 · 2012 · Paper 2Calculate an unbiased estimate for the standard deviation of the lifetime of the battery of the computer.
  25. 5(a)(i)3 marks· CAPE Applied Mathematics Unit 1 · 2013 · Paper 2State the distribution of the sample mean, X̄, giving the value of its parameters.
  26. 5(a)(ii)5 marks· CAPE Applied Mathematics Unit 1 · 2013 · Paper 2Calculate P(X̄ ≥ 38.7).
  27. 5(b)(i)4 marks· CAPE Applied Mathematics Unit 1 · 2013 · Paper 2Calculate a 94 per cent confidence interval for the true mean mass, µ, of the packages.
  28. 5(b)(ii)2 marks· CAPE Applied Mathematics Unit 1 · 2013 · Paper 240 random samples of the packages of biscuits were taken, and a 94 per cent confidence interval for µ was found for each. Determine the expected number of intervals that will contain µ.
  29. Q381 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2014 · Paper 1Which of the following conditions are satisfied for the use of the central limit theorem?
  30. Q411 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2014 · Paper 1Given a random sample of size n with mean \bar{x} and standard deviation \hat{\sigma}, then a 90\% confidence interval for the population mean is
  31. Q421 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2014 · Paper 1Two hundred samples are chosen and 90% confidence intervals calculated for each. How many of these samples are NOT expected to contain the population mean? The confidence level used is
  32. Q451 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2014 · Paper 1The mass of a bag of potatoes is normally distributed with mean mass 50\text{ kg} and standard deviation 6\text{ kg}. Five bags of potatoes are chosen at random. Given…
  33. 5(a)(i)a)3 marks· CAPE Applied Mathematics Unit 1 · 2014 · Paper 2Calculate an unbiased estimator for the mean number of minutes students arrive late.
  34. 5(a)(i)b)4 marks· CAPE Applied Mathematics Unit 1 · 2014 · Paper 2Calculate an unbiased estimator for the variance of the minutes students arrive late.
  35. 5(b)6 marks· CAPE Applied Mathematics Unit 1 · 2014 · Paper 2In a sample of 52 students, 18 arrived late for class. Construct a 95% confidence interval for the population proportion of students who arrive late.
  36. Q331 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2015 · Paper 1If samples of size n are drawn from a population which is not normally distributed, then the sampling distribution of the sample mean is
  37. Q441 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2015 · Paper 1The unbiased estimate for the proportion of students who scored less than 20 is
  38. Q451 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2015 · Paper 1To construct a confidence interval for the population proportion, p, which of the following must be true for a sample of size n?
  39. 5(a)(i)2 marks· CAPE Applied Mathematics Unit 1 · 2015 · Paper 2Calculate the width of the interval.
  40. 5(a)(ii)2 marks· CAPE Applied Mathematics Unit 1 · 2015 · Paper 2Calculate the mean of the sample.
  41. 5(a)(iii)3 marks· CAPE Applied Mathematics Unit 1 · 2015 · Paper 2Calculate the standard deviation of the value of the sales invoice.
  42. 5(b)(i)3 marks· CAPE Applied Mathematics Unit 1 · 2015 · Paper 2State the approximate distribution modelled by X̄, giving its parameters.
  43. 5(b)(ii)5 marks· CAPE Applied Mathematics Unit 1 · 2015 · Paper 2Calculate P(X̄ < 28).
  44. Q321 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2016 · Paper 1Which of the following conditions will widen the confidence interval for a population mean \mu? I. The level of significance increases. II. The sample size increases. III. The variance increases.
  45. Q381 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2016 · Paper 1Which of the following conditions are appropriate for the use of the central limit theorem?
  46. Q431 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2016 · Paper 1The weights of iron rods manufactured by a certain company are normally distributed with a standard deviation of 3.4\text{ kg}. A random sample of 35 iron rods with mean weight of 40.2\text{ kg} is selected. A 99%…
  47. Q451 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2016 · Paper 1The mass of a bag of potatoes is normally distributed with mean mass 50\text{ kg} and standard deviation 6\text{ kg}. Five bags of potatoes are chosen at random. Given \bar{X} \sim N\left(50, \frac{36}{5}\right),…
  48. 5(a)(i)2 marks· CAPE Applied Mathematics Unit 1 · 2016 · Paper 2Calculate unbiased estimates for the population mean, µ.
  49. 5(a)(ii)3 marks· CAPE Applied Mathematics Unit 1 · 2016 · Paper 2Calculate unbiased estimates for the standard deviation, σ, of the length of time between planting and germination.
  50. 5(a)(iii)3 marks· CAPE Applied Mathematics Unit 1 · 2016 · Paper 2Construct a 94% confidence interval for the mean length of time it takes for this type of pepper seed to germinate.
  51. 5(a)(iv)2 marks· CAPE Applied Mathematics Unit 1 · 2016 · Paper 2Sixty random samples of 49 pepper seeds are taken and a 95% confidence interval for µ is found for each sample. Determine the appropriate number of intervals that will contain the population mean, µ.
  52. 5(b)5 marks· CAPE Applied Mathematics Unit 1 · 2016 · Paper 2The mean of 150 observations of X, where X ~ Bin(135, 0.36) is X̄. Use the central limit theorem to state the distribution that is modelled by X̄, giving the values of its parameters.
  53. Q401 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2017 · Paper 1When 60 random samples of size 45 are taken, a 90\% confidence interval for the population mean, \mu, is found for each sample. The expected number of samples that do NOT contain \mu is
  54. Q441 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2017 · Paper 1Item 44 refers to the following information which shows the scores obtained by a sample of 40 students in a class quiz. 25 31 33 17 20 19 39 32 33 12 40 22 25 23 29 15 15 17 27 40 31 35 26 19 29 29 39 33 35 37 32 31 21…
  55. Q451 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2017 · Paper 1To construct a confidence interval for the population proportion, p, which of the following must be true for a sample of size n?
  56. 5(a)4 marks· CAPE Applied Mathematics Unit 1 · 2017 · Paper 2A random sample of 55 sweets was found to have a mean mass of 0.93 g and a standard deviation of 0.10 g. Determine an approximate 99% confidence interval for the mean mass of the sweets.
  57. 5(b)4 marks· CAPE Applied Mathematics Unit 1 · 2017 · Paper 2A certain supermarket found that the price codes on some packages were not clear. In a random sample of 80 packages, 18 had price codes that were not clear. Construct a 97% confidence interval for the proportion of…
  58. 5(c)6 marks· CAPE Applied Mathematics Unit 1 · 2017 · Paper 2A random variable X follows a normal distribution with mean, μ, and an unknown variance, σ². A random sample of 150 observations of X gave ∑x = 1600 and ∑x² = 18 040. Calculate unbiased estimates for the mean and…
  59. 5(d)2 marks· CAPE Applied Mathematics Unit 1 · 2017 · Paper 2Forty random samples of 5 soap bars are taken and a 90% confidence interval for the mean mass, μ, is calculated for EACH sample. Find the expected number of intervals that do NOT contain μ.
  60. 5(e)(i)4 marks· CAPE Applied Mathematics Unit 1 · 2017 · Paper 2State the distribution of X̄, giving its parameters.
  61. 5(e)(ii)5 marks· CAPE Applied Mathematics Unit 1 · 2017 · Paper 2Calculate P(X̄ > 4.75).
  62. Q311 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2018 · Paper 1A random sample of 81 items produced a 95% confidence interval for the population mean of 573.08. If 40 such samples were examined, how many of them are expected to contain a true population mean?
  63. Q381 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2018 · Paper 1Which of the following conditions are satisfied for the use of the central limit theorem?
  64. Q431 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2018 · Paper 1The weights of iron rods manufactured by a certain company are normally distributed with a standard deviation of 3.4 kg. A random sample of 35 iron rods with mean weight of 40.2 kg is selected. A 99% confidence interval…
  65. Q451 mark · multiple choice· CAPE Applied Mathematics Unit 1 · 2018 · Paper 1To construct a confidence interval for the population proportion, p, which of the following must be true for a sample of size n?
  66. 5(a)(i)5 marks· CAPE Applied Mathematics Unit 1 · 2021 · Paper 2Calculate unbiased estimates for the mean and standard deviation of X.
  67. 5(a)(ii)2 marks· CAPE Applied Mathematics Unit 1 · 2021 · Paper 2State an appropriate distribution for the sample mean, X-bar.
  68. 5(a)(iii)3 marks· CAPE Applied Mathematics Unit 1 · 2021 · Paper 2Determine an approximate 95% confidence interval for the unknown population mean, mu.
  69. 6(b)4 marks· CAPE Applied Mathematics Unit 1 · 2021 · Paper 2A random sample of 384 provided a 95% confidence interval for a population proportion, p, with a margin of error of 0.05. Approximate the value of p based on this information.
  70. 5(a)(ii)2 marks· CAPE Applied Mathematics Unit 1 · 2022 · Paper 2Determine if the required conditions for the construction of a confidence interval for p are satisfied.
  71. 5(a)(iii)3 marks· CAPE Applied Mathematics Unit 1 · 2022 · Paper 2Identify the sampling distribution of p-hat. You must state the specific parameters of the sampling distribution.
  72. 5(a)(iv)4 marks· CAPE Applied Mathematics Unit 1 · 2022 · Paper 2Construct a two-sided 90% confidence interval for p.
  73. 5(a)(v)2 marks· CAPE Applied Mathematics Unit 1 · 2022 · Paper 2Interpret the obtained confidence interval in the context of the problem.
  74. 5(b)5 marks· CAPE Applied Mathematics Unit 1 · 2022 · Paper 2A scientist collected a random sample of 100 measurements from a normal population with a mean of 500 and a standard deviation of 80. Determine the probability that the scientist will get a sample mean within the values…

Related topics in Module 3: Analysing and Interpreting Data