26134 · Responsible Evidence-based Decisions
Insights from Limited Information: Sampling Distributions
Week 6 opens Module 2 with the key idea behind all inference: a sample statistic is itself a random variable. You meet the sampling distribution of the sample mean, its standard error σ/√n, and the Central Limit Theorem that makes the sample mean approximately normal for large n. This underpins every confidence interval and hypothesis test to come, and exam items ask for a probability about a sample mean.
What this chapter covers
- 01Population parameter (μ, σ, p — fixed, unknown) vs sample statistic (x̄, s, p̂ — random)
- 02A statistic is a random variable: it varies from sample to sample
- 03Sampling distribution of the sample mean: E(x̄) = μ
- 04Standard error of the mean SE = σ/√n and why it shrinks as n grows
- 05Central Limit Theorem: x̄ is approximately normal for large n regardless of population shape (rule of thumb n ≥ 30)
- 06Standardising the sample mean: Z = (x̄ − μ)/(σ/√n) when σ is known
- 07Unknown σ → t with n − 1 df; sample variance → chi-square with n − 1 df
A probability about the sample mean using the CLT
- +1Sampling distribution. E(x̄) = μ = 500 and, by the Central Limit Theorem, since n = 64 ≥ 30 the sample mean is approximately normal even if the population is not.
- +1Standard error. SE = σ/√n = 80/√64 = 80/8 = 10. Note this is far tighter than the population SD of 80 — averaging over 64 observations reduces variability.
- +1Standardise. z = (x̄ − μ)/SE = (520 − 500)/10 = 20/10 = 2.0.
- +1Tail probability. P(x̄ > 520) = P(Z > 2.0) = 1 − Φ(2.0) = 1 − 0.9772 = 0.0228 (about 2.3%).
Key terms
- Parameter vs statistic
- A population parameter (μ, σ, p) is a fixed but unknown feature of the whole population; a sample statistic (x̄, s, p̂) is computed from a sample and varies from sample to sample. Inference uses the random statistic to estimate the fixed parameter.
- Sampling distribution
- The distribution of a statistic (such as x̄) across all possible samples of size n. It lets you assign probabilities to values of the statistic, for example P(x̄ > 520).
- Standard error (of the mean)
- The standard deviation of the sample mean's sampling distribution, SE = σ/√n (or s/√n when σ is unknown). It quantifies how much x̄ bounces around μ and shrinks as n grows.
- Central Limit Theorem (CLT)
- For a sufficiently large sample size (rule of thumb n ≥ 30), the sampling distribution of x̄ is approximately normal with mean μ and standard error σ/√n, regardless of the population's shape. If the population is itself normal, x̄ is exactly normal for any n.
- Standardised sample mean
- Z = (x̄ − μ)/(σ/√n) follows N(0, 1) when σ is known, letting you find probabilities about x̄ from the Z-table. When σ is unknown, replace σ with s and use t with n − 1 degrees of freedom.
- Sampling distribution of the variance
- For a normal population, (n − 1)s²/σ² follows a chi-square distribution with n − 1 degrees of freedom, which is right-skewed and non-negative. It underlies inference about a variance or standard deviation.
Insights from Limited Information: Sampling Distributions FAQ
What is the difference between the standard deviation and the standard error?
The standard deviation σ describes the spread of individual observations in the population. The standard error σ/√n describes the spread of the SAMPLE MEAN across repeated samples. The SE is smaller by a factor of √n, which is why averaging many observations gives a much more stable estimate.
What does the Central Limit Theorem let me do?
It lets you treat the sample mean as approximately normal once n is reasonably large (commonly n ≥ 30), even when the underlying population is skewed or non-normal. That is what makes Z-based (and t-based) confidence intervals and hypothesis tests about a mean valid in practice.
When do I use σ/√n instead of σ?
Use σ when the question is about a single observation X; use the standard error σ/√n when it is about a sample mean x̄. A very common exam error is standardising a sample mean with σ alone — that ignores the averaging and gives far too large a tail probability.
Can AI help me with sampling distributions in 26134?
Yes, as a study aid. Sia can explain why a statistic is a random variable, walk you through computing a standard error and a probability about a sample mean, and show when the CLT applies, step by step. Use it to rehearse the method; it does not do your graded work, and the UTS academic-integrity policy applies.
Exam move
This is the conceptual pivot of the course, so make sure you can say in one line why a sample mean is a random variable and what the CLT gives you. Drill the standard-error calculation SE = σ/√n and the standardisation Z = (x̄ − μ)/SE until automatic, and always check whether a question is about a single value (σ) or a mean (σ/√n). Rehearse the three companion results: σ known → Z, σ unknown → t with n − 1 df, variance → chi-square with n − 1 df, because they set up Weeks 7–9. Keep the SE and standardisation formulas on your printed notes. When the parameter-versus-statistic distinction blurs, ask Sia to re-explain it on a fresh example; confirm assessment details on Canvas.
Working through Insights from Limited Information: Sampling Distributions in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Insights from Limited Information: Sampling Distributions question and get a clear, step-by-step explanation grounded in how 26134 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
