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NSWMaths Standard 2Quick questions

Year 12: Statistical Analysis

Quick questions on z-scores, standardisation and comparing normal distributions for HSC Maths Standard 2

11short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the z-score formula?
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For a value xx from a normal distribution with mean μ\mu and standard deviation σ\sigma:
What is reverse?
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Rearrange to find xx given a z-score:
What is z-scores and percentiles?
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Common percentile-to-z-score values (from the standard normal table):
What is z-scores and probabilities?
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To find the probability that an observation is less than some value xx:
What is cross-distribution comparison?
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A Sydney accountant earns \135000inanindustrywherethemeanis in an industry where the mean is \110000110000 and SD is \20000.AMelbourneconsultantearns. A Melbourne consultant earns \120000120000 in an industry where the mean is \95000andSDis and SD is \1500015000.
What is finding a percentile?
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For exam marks normally distributed with μ=65\mu = 65, σ=12\sigma = 12, find the cutoff for the top 5%5\%.
What is australian salary example (ABS-style 2024 data)?
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Full-time adult average weekly earnings (ABS, August 2024) are approximately \1830.AssumenormalwithSD. Assume normal with SD \500500.
What is confusing σ\sigma and σ2\sigma^2?
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Use the standard deviation, not the variance.
What is forgetting to take both directions?
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P(∣Z∣>1)P(|Z| > 1) is both tails: 2×0.16=32%2 \times 0.16 = 32\%. Not just 16%16\%.
What is reading the table wrong?
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Φ(z)\Phi(z) is P(Z≤z)P(Z \le z). For "greater than", use 1−Φ(z)1 - \Phi(z).
What is using z-scores on non-normal data?
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Z-scores are most useful for normal distributions. The empirical rule percentages do not apply to skewed data. :::

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