ETX5900 · Business Statistics
The Normal Distribution & Sampling Distributions
This is Module 4 (Week 4) of ETX5900 Business Statistics at Monash University, the postgraduate business-statistics unit run by the Department of Econometrics & Business Statistics (textbook: Berenson et al., Chapters 6-7). It turns the normal model into a working tool: standardise any value to a z-score, read the cumulative standard-normal table, and understand why a sample mean has its own distribution with standard error σ/√n.
The Central Limit Theorem covered here is the foundation every later inference relies on - confidence intervals in Module 5 and hypothesis tests in Modules 7-8 - so it is heavily represented in the exam.
What this chapter covers
- 01The normal model X ~ N(μ, σ²) and standardising to Z = (X − μ)/σ ~ N(0,1)
- 02Reading the cumulative standard-normal table: left = Φ(z), right = 1 − Φ(z), between = Φ(b) − Φ(a)
- 03The empirical 68-95-99.7 rule and key z landmarks (1.645, 1.960, 2.576; 90th percentile z = 1.28)
- 04Inverse-normal problems: finding a value or percentile from an area via X = μ + zσ
- 05The sampling distribution of the sample mean x̄: mean = μ, standard error = σ/√n
- 06The Central Limit Theorem: x̄ is approximately normal for n ≥ 30 whatever the population shape
- 07Standard deviation σ (spread of X) vs standard error σ/√n (spread of x̄)
- 08The sampling distribution of a proportion p: mean π, SE = √(π(1−π)/n)
- 09Foundations of inference: z (σ known) vs t (σ unknown, df = n − 1) and the reject / fail-to-reject logic
z-test for a mean when the population standard deviation is known
- +1State the hypotheses. Because the claim to test is 'exceeds 45', this is a one-tailed (upper) test: H0: μ = 45 versus H1: μ > 45.
- +1Choose the statistic. The population σ is KNOWN, so use the z-test (normal), not t.
- +1Compute the standard error and z. SE = σ/√n = 10/√100 = 10/10 = 1.0, so z = (47.5 − 45)/1.0 = 2.50.
- +1Find the critical value / p-value. One-tailed critical z at α = 0.05 is 1.645; the p-value is P(Z > 2.50) = 1 − Φ(2.50) = 1 − 0.9938 = 0.0062.
- +1Decide and conclude in context. Since 2.50 > 1.645 (and 0.0062 < 0.05), REJECT H0: at the 5% level there is sufficient evidence that the mean checkout time exceeds 45 seconds.
Key terms
- Standardising / z-score
- Re-expressing a value as z = (X − μ)/σ, the number of standard deviations it lies from the mean, so one shared standard-normal table serves every normal variable.
- Cumulative standard-normal table
- A table giving Φ(z) = P(Z ≤ z), the area to the LEFT of z. Right-tail and between-values probabilities are obtained by subtraction from it.
- Sampling distribution of x̄
- The distribution of the sample mean over all possible samples of size n: it is centred on the population mean μ with standard deviation equal to the standard error σ/√n.
- Standard error (SE)
- The standard deviation of a sample statistic. For the mean, SE = σ/√n; it is smaller than σ and shrinks as the sample size n increases.
- Central Limit Theorem (CLT)
- For a large enough sample (rule of thumb n ≥ 30), the sampling distribution of x̄ is approximately normal regardless of the population's shape.
- z (σ known) vs t (σ unknown)
- Standardise a mean with z = (x̄ − μ)/(σ/√n) when σ is known; use the Student-t statistic t = (x̄ − μ)/(s/√n) with df = n − 1 when σ is estimated by the sample s.
- Reject vs fail to reject H0
- In a test you either reject H0 (the statistic falls beyond the critical value, or p < α) or fail to reject it - you never 'accept' H0 or call it true.
- Empirical (68-95-99.7) rule
- For bell-shaped data, about 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean respectively.
The Normal Distribution & Sampling Distributions FAQ
When do I divide by √n and when do I not?
Divide by √n (use the standard error σ/√n) whenever the question is about a SAMPLE MEAN or average of n observations. Use σ on its own only when the question is about a single individual value X. Mixing these up is the most common lost mark in this module.
Do I use z or t for a mean?
Use z (the normal) when the population standard deviation σ is known, and the Student-t with df = n − 1 when σ is unknown and estimated by the sample standard deviation s. In real business data σ is usually unknown, so t is the practical default - a distinction examined from Module 5 onward.
Can AI help me with the normal distribution and sampling distributions in ETX5900?
Yes - Sia can explain these ideas step by step: how to standardise to a z-score, which direction to read the cumulative table, why the standard error is σ/√n, and what the Central Limit Theorem lets you assume. It walks you through the method and checks your reasoning on practice questions, but it does not sit your assessments or guarantee any grade - the aim is to help you understand the technique so you can apply it yourself and confirm expectations on Moodle.
Exam move
Master four objects and the moves between them: a raw value X, its z-score, a sample mean x̄, and the standard error σ/√n that scales it. Drill the three table directions (left = Φ(z), right = 1 − Φ(z), between = Φ(b) − Φ(a)) until they are automatic, and always sketch the bell and shade the area you actually want before reading the table - that single habit prevents the wrong-tail error. Practise spotting the 'single value vs average' cue in the wording, since that decides whether you divide by σ or by σ/√n. The exam provides a formula sheet and statistical tables, so rehearse choosing and reading them rather than memorising, carry four decimal places in probabilities, and phrase every conclusion in context. Because this chapter is the foundation for confidence intervals and hypothesis tests, time spent here compounds across the rest of the unit; spend revision time in proportion to marks and confirm all exam details on Moodle.
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