26134 · Responsible Evidence-based Decisions
Continuous Outcomes: The Normal Distribution
Week 5 moves to continuous random variables, where probability is area under a density curve, and to the normal distribution in particular. You standardise to Z-scores and use the Z-table to find tail and interval probabilities. This standard normal is the engine behind the confidence intervals and hypothesis tests that follow, and Z-table lookups are a guaranteed exam item.
What this chapter covers
- 01Continuous random variable and probability density function f(x): probability = area, total area = 1
- 02Why P(X = an exact value) = 0 for a continuous variable
- 03The normal distribution N(μ, σ²): bell-shaped, symmetric, mean = median = mode
- 04The empirical 68–95–99.7 rule
- 05Standardisation to a Z-score: Z = (X − μ)/σ, giving Z ~ N(0, 1)
- 06Reading the Z-table for P(Z ≤ z), and using symmetry for lower and upper tails
- 07Interval probability P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
- 08Excel: NORM.DIST, NORM.S.DIST, NORM.S.INV
Normal probabilities via the Z-table
- +1Standardise 620. z = (620 − 500)/80 = 120/80 = 1.5.
- +1Upper tail. P(X > 620) = P(Z > 1.5) = 1 − Φ(1.5) = 1 − 0.9332 = 0.0668 (about 6.7%).
- +1Standardise 440. z = (440 − 500)/80 = −60/80 = −0.75. By symmetry Φ(−0.75) = 1 − Φ(0.75) = 1 − 0.7734 = 0.2266.
- +1Interval probability. P(440 < X < 620) = Φ(1.5) − Φ(−0.75) = 0.9332 − 0.2266 = 0.7066 (about 70.7%).
Key terms
- Probability density function (PDF)
- For a continuous random variable, f(x) ≥ 0 describes relative likelihood, and probability is the AREA under the curve over an interval, with total area 1. A single point has zero width, so P(X = a) = 0.
- Normal distribution
- A symmetric, bell-shaped distribution N(μ, σ²) with location μ and spread σ, where mean = median = mode. About 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean (the empirical rule).
- Standardisation / Z-score
- Z = (X − μ)/σ converts any normal value to the standard normal N(0, 1), measuring how many standard deviations X sits from its mean. It lets one Z-table serve every normal distribution.
- Standard normal (Z) distribution
- The normal with mean 0 and SD 1. The Z-table gives P(Z ≤ z) = Φ(z), the area to the left of z; upper tails use 1 − Φ(z) and negative values use the symmetry Φ(−z) = 1 − Φ(z).
- Empirical (68–95–99.7) rule
- For a normal distribution, roughly 68% of values fall within μ ± 1σ, 95% within μ ± 2σ and 99.7% within μ ± 3σ. A quick sanity check for whether a probability answer is plausible.
- Interval probability
- For a continuous variable, P(a ≤ X ≤ b) = F(b) − F(a), the difference of two cumulative (left-tail) probabilities. In standardised form it is Φ(z_b) − Φ(z_a).
Continuous Outcomes: The Normal Distribution FAQ
Why is the probability of an exact value zero for a continuous variable?
Because probability is area under the density curve, and a single point has zero width, hence zero area. So P(X = 25.0) = 0 for a continuous variable; only intervals like P(24.5 < X < 25.5) carry positive probability. This is why continuous questions always ask about ranges.
How do I use the Z-table for an upper tail or a negative z?
The table gives Φ(z) = P(Z ≤ z), the area to the left. For an upper tail P(Z > z) = 1 − Φ(z). For a negative value use symmetry: Φ(−z) = 1 − Φ(z). Always sketch the bell and shade the target area first, then decide whether to subtract from 1 or take a difference.
What does standardising actually do?
It re-expresses a value as a number of standard deviations from the mean, Z = (X − μ)/σ, converting any normal distribution to the single standard normal N(0, 1). That means one Z-table works for every problem, and a z-score of 1.5 has the same tail probability whatever μ and σ are.
Can AI help me with the normal distribution in 26134?
Yes, as a study aid. Sia can walk you through standardising a value, reading Φ(z) off the Z-table, handling upper tails and intervals, and sketching the shaded region, one step at a time. Use it to rehearse; it does not do your graded assessment, and the UTS academic-integrity policy applies.
Exam move
The Z-table lookup is a guaranteed exam item, so make it reflexive. Practise the three cases until automatic: left tail Φ(z), upper tail 1 − Φ(z), and interval Φ(z_b) − Φ(z_a), always sketching the bell and shading first. Rehearse negative z using symmetry, and keep the empirical 68–95–99.7 rule handy as a plausibility check. Because the exam supplies the Z-table, drill reading it accurately rather than memorising values, and note the Excel equivalents (NORM.S.DIST, NORM.S.INV) for the open-book setting. Keep the standardisation formula and a labelled bell curve on your printed notes. When a lookup confuses you, ask Sia to work a fresh N(μ, σ²) problem with you; confirm assessment details on Canvas.
Working through Continuous Outcomes: The Normal Distribution in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Continuous Outcomes: The Normal Distribution 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.
