Activity 25

MATH 216: Statistical Thinking

One-Sample Inference Activity: t-test & z-test

Review & Flowchart

Key concepts:

• Confidence Interval (CI):

  \(\bar{x} \pm t_{\frac{\alpha}{2}, n-1} \frac{s}{\sqrt{n}}\) (unknown \(\sigma\))
  \(\bar{x} \pm z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}\) (known \(\sigma\))

Hypothesis test flow:

graph TD
  A[Start] --> B{σ known?}
  B -->|Yes| C["Use z-test/z-interval"]
  B -->|No| D{n ≥ 30?}
  D -->|Yes| E["CLT: Use t-test (z ≈ t)"]
  D -->|No| F["Normal? QQ-plot/test"]
  F -->|Yes| G[Use t-test]
  F -->|No| H[Non-parametric test]

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Activity 1: CI for Small Sample (Unknown σ)

Problem: Construct 95% CI for blood pressure increase (n=6).

Data: 1.7, 3.0, 0.8, 3.4, 2.7, 2.1

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Activity 2: Hypothesis Test (Unknown σ, Moderate n)

Problem: Test if pH ≠ 8.5 (α=0.05, n=17, \(\bar{x}=8.42\), \(s=0.16\)).

Activity 3: Known σ (z-test)

Problem: Bolts have σ=0.15 cm. Sample (n=20): \(\bar{x}=5.08\) cm. Test if μ≠5.

Activity 4: Large Sample Approximation

Problem: Commute time (n=50, \(\bar{x}=35\), s=8). Test if \(\mu \neq 30\).
Tasks: