Math, worked out.

Explanation

Statistics – Descriptive Statistics Chapter Map

Descriptive statistics: population vs sample; measures of centre (mean, median, mode); dispersion (range, SD); relative position (Z-score, percentiles).

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Statistics – Point vs Interval Estimation

Point estimation gives one value (e.g. x̄ = 75); interval estimation gives a range (e.g. [72,78] at 95% confidence); advantages of interval over point.

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Statistics – Confidence Level and Significance Level

Confidence level (1−α) expresses % certainty; significance level α is its complement; table of Z-critical values: 1.645, 1.96, 2.576 for 90/95/99%.

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Statistics – Confidence Interval for Mean (Known Variance)

Known-variance CI: X̄ ± Z_{α/2}·σ/√n; Z_{α/2}=1.96 for 95%; components: sample mean, Z-critical, population SD, sample size; worked example.

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Statistics – Confidence Interval for Mean (Unknown Variance)

CI for mean (unknown variance): X̄ ± t_{α/2}·S/√n using t-distribution with df=n−1; t is wider than Z to account for extra uncertainty; t→Z for large n.

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Statistics – Confidence Interval for a Proportion

Proportion CI: p̂ ± Z_{α/2}·√(p̂(1−p̂)/n); requires np̂≥5 and n(1−p̂)≥5; full worked example: 80 successes from n=200, p̂=0.4, 95% CI=[0.332,0.468].

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