Week 4 (biostats) - Sampling Distribution & CLT

Central Limit Theorem

• For large n, the distribution of the sample mean is normal with mean $\mu$ and standard deviation $SE = \frac{\sigma}{\sqrt{n}}$, even if the population distribution is non-normal.

Sampling Distribution

• Repeated independent samples of size $n$ from the population; compute the sample mean for each; the histogram of these sample means is the sampling distribution.
• In data: weight distribution appears normal; pre-op creatinine is right-skewed.

Standard Error (SE)

• Measures dispersion of the sampling distribution; reflects precision of the sample mean as estimator of the population mean.
• Technical: $SE = \frac{SD}{\sqrt{n}}$ when using population SD.
• If only sample SD is available: $SE \approx \frac{S}{\sqrt{n}}$.

Relationship SE, SD, and n

• General formula: $SE = \frac{SD}{\sqrt{n}}$.
• As n increases, SE decreases; larger SD increases SE.

68-95-99.7% Rule for sampling distribution

• 68% of sample means lie within $\mu \pm SE$.
• 95% lie within $\mu \pm 2SE$.
• 99.7% lie within $\mu \pm 3SE$.

Normal vs T distributions; Z-score and T-score

• If population SD known: Z-score, $Z = \frac{\bar{X} - \mu}{SE}$ with $SE = \frac{\sigma}{\sqrt{n}}$.
• If population SD unknown: use T-score, $t = \frac{\bar{X} - \mu}{SE}$ with $SE \approx \frac{S}{\sqrt{n}}$ and degrees of freedom $df = n - 1$.
• As df increases, t-distribution approaches normal.

Worked Example 1 – Birth weight

• Given: $\mu = 112\,\text{oz}, \sigma = 20.6\,\text{oz}, n = 100.$
• Range: between $107.571$ and $116.429$ oz for the sample mean.
• SE: $SE = \frac{20.6}{\sqrt{100}} = 2.06$.
• Z bounds: $Z<em>1 = \frac{107.571 - 112}{2.06} \approx -2.15, \quad Z</em>2 = \frac{116.429 - 112}{2.06} \approx 2.15$.
• Probability: $P(107.571 \le \bar{X} \le 116.429) \approx 0.9684 \ (96.84\%)$.

Worked Example 2 – Birth weight

• Given: $\mu = 112\,\text{oz}, n = 10, S = 20.6\,\text{oz}$.
• SE: $SE \approx \frac{S}{\sqrt{n}} = \frac{20.6}{\sqrt{10}} \approx 6.51$.
• Target: P(\bar{X} > 121).
• t-score: $t = \frac{121 - 112}{6.51} \approx 1.38$; df = 9.
• Look up p-value: one-tailed p ≈ 0.10.
• Probability: ≈ 10%.

Degrees of Freedom

• df = number of observations that can vary when estimating a parameter.
• For t-tests with single sample SD: df = n - 1.
• Example: df = 9 for n = 10; use appropriate t-table value; for df > 120, approximate with infinite df.

Practical recap: When to use SD vs SE

• Describe spread of observed data: use SD.
• Infer population mean from sample: use SE (and Z or T as appropriate).

Summary

• Central Limit Theorem underpins normality of the sampling distribution of the mean for large n.
• The sampling distribution of the mean has mean $\mu$ and SE $\frac{\sigma}{\sqrt{n}}$ (or estimated with $\frac{S}{\sqrt{n}}$).
• Use Z when $\sigma$ known; use T when $\sigma$ unknown; df = n-1.
• SE decreases as n grows; larger SD increases SE.
• The CLT supports inference on the mean using the normal model for the sampling distribution.