H2 Maths Notes (JC 1-2): 6.4) Sampling

Before you revise\ Recap descriptive statistics (mean, variance) so you can translate between population parameters and sample statistics. Keep a table of common critical values (\( z_{0.95} = 1.645 \), \( z_{0.975} = 1.96 \), t-values for small \( n \)) beside you while practising.

Sampling Language

• A population has true mean \( \mu \) and variance \( \sigma^2 \).
• A sample of size \( n \) yields statistics \( \bar{X} \) (sample mean) and \( S^2 \) (sample variance).
• Sampling methods: simple random, stratified, systematic, cluster. Be ready to describe each procedure in words and identify bias sources.

Distribution of the Sample Mean

• If \( X_1, X_2, \dots, X_n \) are independent with mean \( \mu \) and variance \( \sigma^2 \), then \( \bar{X} = \frac{1}{n} \sum X_i \) has \( E(\bar{X}) = \mu \) and \( \operatorname{Var}(\bar{X}) = \frac{\sigma^2}{n} \).
• For large \( n \) (Central Limit Theorem), \( \bar{X} \) is approximately \( \mathcal{N}(\mu, \sigma^2/n) \) even if the parent distribution is non-normal.
• If the population is normal, \( \bar{X} \) is exactly normal for any \( n \).

Example -- Average revision hours

JC students have revision hours with \( \mu = 6.4 \), \( \sigma = 1.8 \). For a random sample of \( n = 36 \), find \( P(\bar{X} > 7) \).

1. \( \bar{X} \sim \mathcal{N}(6.4, 1.8^2 / 36) = \mathcal{N}(6.4, 0.09) \).
2. Standardise: \( z = \frac{7 - 6.4}{0.3} = 2.0 \).
3. Probability \( P(\bar{X} > 7) = 1 - \Phi(2.0) \approx 0.0228 \).

Confidence Intervals

• For known \( \sigma \) or large \( n \), a \( (1 - \alpha) \) confidence interval for \( \mu \) is \( \bar{X} \pm z_{1 - \alpha/2} \frac{\sigma}{\sqrt{n}} \).
• When \( \sigma \) is unknown and \( n \leq 30 \), use the t-distribution with \( n - 1 \) degrees of freedom: \( \bar{X} \pm t_{1 - \alpha/2, n-1} \frac{S}{\sqrt{n}} \).

Example -- Interval for a mean

A pilot sample of \( n = 25 \) chemistry students gives \( \bar{X} = 12.4 \) hours of weekly study with sample standard deviation \( s = 2.1 \). Construct a 95% confidence interval for \( \mu \).

1. Small sample, unknown \( \sigma \) \(\Rightarrow\) t-interval with \( 24 \) degrees of freedom.
2. \( t_{0.975, 24} = 2.064 \).
3. Margin \( = 2.064 \times \frac{2.1}{\sqrt{25}} = 0.867 \).
4. Interval: \( (11.5, 13.3) \) hours.

Sampling Proportions

• For \( n \) large, \( \hat{p} = \frac{X}{n} \) (sample proportion) satisfies \( \hat{p} \approx \mathcal{N}\left(p, \frac{p(1 - p)}{n}\right) \).
• Confidence interval: \( \hat{p} \pm z_{1 - \alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \).
• Check \( np \) and \( n(1 - p) \) \( \geq 5 \) before using the approximation.

Example -- Device adoption

In a survey of \( 180 \) JC students, \( 126 \) use an iPad for note-taking. Estimate the true proportion with 90% confidence.

1. \( \hat{p} = \frac{126}{180} = 0.700 \).
2. \( z_{0.95} = 1.645 \).
3. Margin \( = 1.645 \sqrt{\frac{0.7 \times 0.3}{180}} = 0.056 \).
4. Interval: \( (0.644, 0.756) \).

Calculator Workflows

• Use GC statistics mode (1-Var Stats) to obtain \( \bar{X} \) and \( S \) from raw data.
• TI: tInterval, zInterval, 1-PropZInt functions automate standard intervals; record the command, parameters, and output range explicitly.
• Casio: Run STAT > DIST > NORM or t for tail probabilities; confidence intervals may need manual computation, so pre-program formulas if allowed.

Exam Watch Points

• State sampling design in words (e.g. “select every 5th student in the register”) and comment on bias.
• Identify whether \( \sigma \) is known and justify the use of \( z \) or \( t \)-intervals.
• Always interpret the interval in context: “We are 95% confident the true mean weekly revision time lies between...”.
• Quote degrees of freedom when using \( t \) and note when CLT assumptions are invoked.

Quick Revision Checklist

• [ ] Distinguish population vs sample parameters quickly (\( \mu \) vs \( \bar{X} \)).
• [ ] Compute \( \operatorname{Var}(\bar{X}) \) and standard errors without mixing up \( S \) and \( \sigma \).
• [ ] Build both mean and proportion confidence intervals, choosing \( z \) or \( t \) correctly.
• [ ] Explain the meaning of a confidence level in sentences, avoiding probability statements about \( \mu \).

Next steps: Combine sampling intervals with hypothesis testing (Topic 6.5) and regression analysis (Topic 6.6) for full Paper 2 long-response practice.