QBUS5001 Chap.6 Sampling Distributions, CLT & Estimation
Sampling Distributions, CLT & Estimation
Modules 5 and 6 connect samples to populations.
The sampling distribution of the sample mean has mean μ and standard error σ/√n; the Central Limit Theorem makes that distribution approximately Normal for n ≥ 30 regardless of the population shape, while the Law of Large Numbers says x̄ converges to μ as n grows.
From there you build confidence intervals (point estimate ± critical value × standard error) for a mean (Z when σ is known, t when it is not) and for a proportion, and you determine the required sample size for a target margin of error.
The interpretation of a CI — a statement about the procedure, not a probability about a fixed parameter — is a perennial exam point.
What this chapter covers
- 01
Sampling distribution of x̄: mean μ, standard error σ/√n
- 02
Central Limit Theorem: approximately Normal for n ≥ 30
- 03
Law of Large Numbers: x̄ → μ as n → ∞
- 04
Sampling distribution of a proportion (np ≥ 5, n(1−p) ≥ 5)
- 05
Unbiased and consistent point estimators
- 06
CI for a mean: σ known → Z, σ unknown → t(n−1)
- 07
CI for a proportion
- 08
Sample-size determination for mean and proportion
- 09
Correct frequentist interpretation of a confidence interval
CLT for the sample mean
- 1 markCheck the CLT applies: n = 144 ≥ 30, so the sampling distribution of x̄ is approximately Normal.
- 1 markState the sampling distribution: x̄ ~ N(12000, 900²/144).
- 1 markCompute the standard error: σ/√n = 900/√144 = 900/12 = 75 hours.
- 1 markStandardise the cut-off: Z = (11880 − 12000)/75 = −120/75 = −1.60.
- 1 markFind the lower-tail probability: P(Z < −1.60) = NORM.S.DIST(−1.60, 1) ≈ 0.0548.
- 1 markConclude: there is about a 5.48% chance the sample mean lifetime falls below 11,880 hours.
Key terms
- Sampling distribution
- The probability distribution of a sample statistic (e.g. x̄) over all possible samples of size n; for the mean it is centred at μ with spread σ/√n.
- Standard error of the mean
- σ/√n, the standard deviation of the sample mean's sampling distribution; it shrinks as n grows, tightening estimates.
- Central Limit Theorem
- For n ≥ 30 the sampling distribution of x̄ is approximately Normal regardless of the population's shape, enabling Normal-based inference on non-Normal data.
- Law of Large Numbers
- As the sample size grows without bound, the sample mean converges to the population mean μ and the standard error tends to zero.
- Unbiased estimator
- An estimator whose expected value equals the parameter it estimates; the sample mean is unbiased because E[x̄] = μ, and it is also consistent by the LLN.
Sampling Distributions, CLT & Estimation FAQ
Should I divide by σ or by σ/√n when standardising a sample mean?
By the standard error σ/√n. A single observation X uses σ; a sample mean x̄ uses σ/√n because averaging reduces variability. Mixing these up is the classic CLT error.
What is the correct interpretation of a 95% confidence interval?
Over repeated sampling, 95% of the intervals constructed this way would contain the true parameter. It is wrong to say there is a 95% probability the parameter lies in this particular interval — the parameter is fixed; it is the interval that is random.
When do I use Z and when t for a confidence interval for a mean?
Use Z (with z(α/2)) when the population standard deviation σ is known. Use t (with t(α/2, n−1)) when σ is unknown and you estimate it with the sample s, which is the usual real-world case.
Exam move
Burn the standard error σ/√n into memory and treat every CLT problem as a Normal probability on x̄. For confidence intervals, memorise the universal template (estimate ± critical × SE) and the Z-vs-t decision (is σ known?), and rehearse the verbal interpretation of a CI word-for-word, because the exam regularly awards a mark purely for the correct frequentist statement.
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