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A Level Maths 9709 · Paper 6 Probability & Statistics 2 · 6.4 · ~4 min read

Sampling and estimation

Populations and random samples, the distribution of the sample mean, the Central Limit Theorem, unbiased estimates of the mean and variance, and confidence intervals for a population mean and a population proportion.

Populations & samplesRandom samplingDistribution of X̄Central Limit TheoremUnbiased estimatess² with n − 1Confidence intervals for μConfidence intervals for p
RSby Riyaz Sir
mentorriyaz.in
Paper tag = where it is examinedFORMULA LIST (MF19) given in the examM1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • X̄ has mean μ and variance σ²/n. Larger samples give means that cluster more tightly around μ.
  • Central Limit Theorem: for large n (n ≥ 30 or so), X̄ is approximately normal whatever the population's distribution.
  • Unbiased estimate of σ²: s² = [Σx² − (Σx)²/n]/(n − 1). Dividing by n − 1 corrects for using x̄ in place of μ.
  • Confidence interval for μ: x̄ ± z × σ/√n. A 95% interval uses z = 1.96; 90% uses 1.645; 99% uses 2.576.

By the end you can

  • Explain populations, samples and random sampling, and why a sample might not be random.
  • Use the distribution of the sample mean, including the Central Limit Theorem.
  • Calculate unbiased estimates of the population mean and variance.
  • Find confidence intervals for a population mean and for a population proportion, and interpret them.

1Samples and the sample mean

X̄ ~ N(μ, σ²/n)

  • Random sample: every possible sample of size n is equally likely to be chosen. Choosing the first 30 customers through the door isn't random.
  • Population normal: X̄ ~ N(μ, σ²/n) exactly, for any n. Population not normal: X̄ ≈ N(μ, σ²/n) for large n (CLT).
405060708090100population: σ = 12X̄ for n = 36: σ/√n = 2Sample means cluster much more tightly around μ, and are approximately normal for large n.
Sample means are much less spread out than single values, and approximately normal for large n.
Worked example

Masses are N(80, 10²). Find the probability that the mean of a random sample of 25 exceeds 83.

  1. X̄ ~ N(80, 100/25 = 4) M1 A1
  2. z = 3/2 = 1.5 ⇒ 1 − 0.9332 = 0.0668 A1
Worked example

A population has mean 5 and variance 4 (not normal). Find the approximate probability that the mean of 40 values is less than 4.6.

  1. by the CLT, X̄ ≈ N(5, 0.1) B1
  2. z = −0.4/√0.1 = −1.265 M1
  3. 0.103 A1
Common mistakeQuoting the CLT for a normal population, or for a small sample from a non-normal one. The CLT is needed only when the population isn't normal, and it needs n large.

2Estimation

Unbiased estimates

x̄ = Σx/n ; s² = [1/(n − 1)][Σx² − (Σx)²/n] FORMULA LIST (MF19)
s² is the unbiased estimate of σ²
Worked example

A sample of 50 has Σx = 1240 and Σx² = 31 200. Find unbiased estimates of μ and σ².

  1. x̄ = 24.8 B1
  2. s² = (31 200 − 1240²/50)/49 = 448/49 M1
  3. = 9.14 A1

Confidence intervals

μ: x̄ ± z σ/√n ; p: p̂ ± z √[p̂(1 − p̂)/n]
use s for σ when n is large and σ is unknown
Worked example

Use the sample above to find a 95% confidence interval for μ.

  1. 24.8 ± 1.96 √(9.143/50) M1
  2. 24.8 ± 0.838 A1
  3. (23.96, 25.64) A1
Worked example

In a random sample of 200 students, 64 use the library weekly. Find a 90% confidence interval for the proportion p.

  1. p̂ = 0.32 B1
  2. 0.32 ± 1.645 √(0.32 × 0.68/200) = 0.32 ± 0.0543 M1 A1
  3. (0.266, 0.374) A1
  • Interpretation: if many such intervals were made, about 95% would contain μ. It doesn't mean "μ is in this interval with probability 0.95".
Common mistakeUsing n instead of n − 1 for s², or σ instead of σ/√n in the interval. Both appear every year in the Paper 6 reports.
Linking questionLinear combinations explain why Var(X̄) = σ²/n: see Linear combinations; the normal is in The normal distribution; and tests on μ are in Hypothesis tests.

Test yourself — cover the answers first

Quick check

  1. X ~ N(10, 9). What is the distribution of the mean of 16 values?
  2. What z-value gives a 99% confidence interval?
  3. Why divide by n − 1 for s²?
  4. When do you need the Central Limit Theorem?
  5. Is "the first 20 names on a list" a random sample?

Exam-style

  1. A sample of 10: Σx = 52, Σx² = 290. Find unbiased estimates of μ and σ².
  2. How large a sample is needed for the 95% interval for μ to have width at most 2, if σ = 6?
  3. A 95% confidence interval for p is (0.22, 0.34). Find p̂ and n.
Show answers
  1. N(10, 9/16)
  2. 2.576
  3. to give an unbiased estimate of σ²
  4. when the population isn't normal and n is large
  5. no
  6. 5.2; (290 − 270.4)/9 = 2.18
  7. 2 × 1.96 × 6/√n ≤ 2 ⇒ √n ≥ 11.76 ⇒ n = 139
  8. p̂ = 0.28; 1.96 √(0.28 × 0.72/n) = 0.06 ⇒ n = 215
Practise this chapter
A Level practice papers
Next chapterHypothesis tests →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by Cambridge University Press & Assessment. Syllabus codes 9709 and 9702 refer to the Cambridge International AS & A Level syllabuses.