When you cannot survey a whole population, statistical estimation uses a single sample to guess the range where the population mean is likely to lie.
Since the sample mean wobbles around the population mean with a spread of σ/√n, you attach a margin of error on either side of that sample mean to build a confidence interval.
Raising the confidence level to 95% makes the z value bigger and widens the interval, while enlarging the sample narrows it, a genuine trade-off.
Move the sample-size n and confidence-level sliders to watch the interval widen and narrow, and note that 95% confidence is not the probability that the population mean lies in this particular interval.
Population & Sample
🔬 What is Estimation?
①When surveying the whole population is impractical
②Estimate the population from a sample
③Key question: how precisely does the sample mean estimate the population mean?
Distribution of Sample Mean
25
10
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How sampling distribution changes with n
Sample Mean Distribution
X̄ ∼ N(μ, σ2n)
For population N(μ,σ²) the sample mean X̄ has this distribution
💡 Magic of Sample Size
①Larger n → narrower distribution of X̄
②Standard error = σ/√n; quadruple n to halve error
③This is the law of large numbers
Confidence Interval
36
95%
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CI: range of plausible μ around the sample mean
CI for Population Mean
x̄ - z × σ√n ≤ μ ≤ x̄ + z × σ√n
Confidence interval for μ when σ is known
📐 Confidence Level and z
①95%: z = 1.96
②99%: z = 2.576
③Higher confidence → wider interval
④Larger n → narrower interval
Meaning of CI
Margin of Error
E = z × σ√n
Half-width of the interval
Required Sample Size
n ≥ (zσE)2
Minimum n for error within E
⚠️ Common Misconception
①"95% CI" does NOT mean "P(μ in this interval) = 95%"
②Correct: of 100 intervals built this way, about 95 contain μ
③μ is fixed; the interval changes from sample to sample
Work It Out
Example 1
From a population with mean m = 100 and SD σ = 20, a sample of size n = 25 is drawn. Find the mean and SD of the sample mean X̄.
1
The mean of the sample mean equals m, and its SD is σ/√n.
E(X̄) = m, σ(X̄) = σ/√n
2
Substitute.
E(X̄) = 100, σ(X̄) = 20/√25 = 4
▸ E(X̄) = 100, σ(X̄) = 4
The SD of the sample mean (standard error) shrinks as the sample grows.
Example 2
From a population with σ = 10, a sample of size n = 25 has sample mean 50. Find the 95% confidence interval for m. (Given P(|Z| ≤ 1.96) = 0.95)
1
Substitute into the confidence-interval formula (σ/√n = 10/5 = 2).
X̄ − 1.96·σ/√n ≤ m ≤ X̄ + 1.96·σ/√n
2
Add and subtract 1.96 × 2 = 3.92.
50 − 3.92 ≤ m ≤ 50 + 3.92, i.e. 46.08 ≤ m ≤ 53.92
▸ 46.08 ≤ m ≤ 53.92
Confidence interval = sample mean ± (confidence coefficient × standard error).
Wrap-up
Core
x̄ - z σ√n ≤ μ ≤ x̄ + z σ√n
Confidence interval for μ (σ known)
CSAT Math (Probability & Statistics) type
From a population with SD σ, a size n = 64 sample gives a 95% confidence interval for the mean of length 4.9. Find σ. (Confidence coefficient 1.96)
①8
②9
③10
④11
⑤12
▸ ③ 10
1
Write the formula for the length of the confidence interval (√64 = 8).