Chapter 1 — Test of Significance
Key Concepts Quick Recap
| Term | Definition |
|---|---|
| Population | Complete set of all possible observations |
| Sample | A subset of the population |
| Parameter | Numerical measure describing a population (μ, σ²) |
| Statistic | Numerical measure calculated from a sample (x̄, s²) |
| Null hypothesis (H₀) | Hypothesis of no difference / no effect |
| Alternative hypothesis (H₁) | Hypothesis that contradicts H₀ |
| Level of significance (α) | Probability of Type I error; commonly 0.05 or 0.01 |
| Critical region | Region of rejection of H₀ |
| Type I error | Rejecting true H₀ (α) |
| Type II error | Accepting false H₀ (β) |
I. Choose the Best Answer (MCQs)
1. A numerical value computed from a sample to estimate a population parameter is called: (a) Parameter (b) Statistic (c) Census (d) Variable
2. The probability of Type I error is denoted by: (a) β (b) 1 − α (c) α (d) 1 − β
| α = P(rejecting H₀ | H₀ is true) = level of significance. |
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3. The probability of Type II error is denoted by: (a) α (b) β (c) 1 − α (d) Power
| β = P(accepting H₀ | H₀ is false). Power of test = 1 − β. |
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4. In a two-tailed test at 5% level of significance, the critical values of z are: (a) ±1.645 (b) ±1.96 (c) ±2.33 (d) ±2.576
5. Which test is used when the population variance is unknown and sample size is small (n < 30)? (a) z-test (b) F-test (c) t-test (d) χ² test
6. Degrees of freedom for a t-test with sample size n is: (a) n (b) n − 1 (c) n + 1 (d) n − 2
7. The null hypothesis H₀: μ = 50 vs H₁: μ ≠ 50 is a: (a) One-tailed (right) test (b) One-tailed (left) test (c) Two-tailed test (d) None
II. Short Answer Questions
Common Mistakes
- Confusing Type I (rejecting true H₀) with Type II (accepting false H₀).
- Using z-test for small samples — use t-test when n < 30 and σ unknown.
- Forgetting that two-tailed z at 5% is ±1.96, not ±1.645 (which is one-tailed).
Exam Checklist
- Parameter (population, Greek letters: μ, σ) vs Statistic (sample, Latin letters: x̄, s).
- α = P(Type I error) = level of significance.
- β = P(Type II error); Power = 1 − β.
- z-test: large samples (n ≥ 30); t-test: small samples (n < 30), unknown σ.
- Two-tailed 5%: z = ±1.96; one-tailed 5%: z = 1.645; two-tailed 1%: z = ±2.576.