BUSS1020 · Quantitative Business Analysis
Sampling Distributions & the CLT
Sampling Distributions & the CLT (Week 6, Berenson Ch 7) explains how a sample statistic behaves across all possible samples — the bridge between describing data and making inferences. You learn that the sample mean X̄ has mean μ and standard error σ/√n, that the sample proportion p has mean π and standard error √[π(1−π)/n], and that the Central Limit Theorem makes both approximately normal for large n regardless of the population's shape. This is the engine behind every confidence interval and hypothesis test that follows.
What this chapter covers
- 01Sampling distribution of a statistic over all possible samples
- 02Distribution of X̄: E(X̄) = μ, standard error σ/√n
- 03Why standard error shrinks as n grows
- 04Finite population correction factor
- 05Central Limit Theorem and its rules of thumb (n ≥ 30, n ≥ 15)
- 06Distribution of the sample proportion p
- 07CLT conditions for proportions: nπ ≥ 5 and n(1−π) ≥ 5
- 08Probabilities about X̄ and p using the normal model
Probability about a sample mean using the CLT
- 1 markThe sampling distribution of X̄ has mean E(X̄) = μ = 2.5 kg.
- 1 markStandard error σ_X̄ = σ/√n = 0.9/√36 = 0.9/6 = 0.15 kg.
- 1 markJustify normality: n = 36 ≥ 30, so by the CLT X̄ is approximately normal even though the population shape is unknown.
- 1 markStandardise the boundary: Z = (2.8 − 2.5)/0.15 = 0.3/0.15 = 2.00.
- 1 markFrom the table P(Z ≤ 2.00) ≈ 0.9772, so P(X̄ > 2.8) = 1 − 0.9772 = 0.0228.
- 1 markConclusion: there is about a 2.3% chance the sample mean exceeds 2.8 kg. Excel: =1 − NORM.DIST(2.8, 2.5, 0.15, 1).
Key terms
- Sampling distribution
- The probability distribution of a sample statistic (such as X̄ or p) computed over all possible samples of a fixed size n.
- Standard error of the mean
- The standard deviation of the sampling distribution of X̄, equal to σ/√n; it measures how much sample means vary and shrinks as n increases.
- Central Limit Theorem (CLT)
- For a sufficiently large sample, the sampling distribution of the sample mean is approximately normal regardless of the population's shape; n ≥ 30 is the usual rule of thumb.
- Standard error of the proportion
- The standard deviation of the sampling distribution of p, equal to √[π(1−π)/n], valid when nπ ≥ 5 and n(1−π) ≥ 5.
- Finite population correction
- The factor √[(N−n)/(N−1)] applied to the standard error when sampling a non-trivial fraction of a finite population, shrinking the standard error.
Sampling Distributions & the CLT FAQ
Why does the population's shape stop mattering for large samples?
Because of the Central Limit Theorem: as n grows, the averaging process smooths out the population's shape and the distribution of X̄ tends toward normal. This is why we can use Z and t methods even when the underlying data are skewed, provided n is large enough.
What sample size counts as 'large enough'?
The common rule of thumb is n ≥ 30 for any population; if the population is already fairly symmetric, n ≥ 15 can be sufficient. For proportions, the conditions are nπ ≥ 5 and n(1−π) ≥ 5.
What's the difference between the standard deviation and the standard error?
The standard deviation σ measures spread among individual observations; the standard error σ/√n measures spread among sample means. The standard error is always smaller (for n > 1) and shrinks further as the sample grows.
Exam move
This week is conceptual glue, so make sure the picture is clear before the arithmetic: a sampling distribution is the distribution of a statistic, its centre equals the parameter, and its spread is the standard error. Drill the σ-versus-σ/√n distinction relentlessly, because it is the single most common error in the back half of the unit. Be able to state the CLT and its conditions in a sentence, and practise both mean and proportion versions. Everything in Weeks 7–9 is just this machinery wrapped in a question, so time invested here compounds.
