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Hypothesis testing; difference of two proportions

Short answer

Hypothesis testing decides whether sample data support rejecting a claim about a population parameter. For two proportions, use pooled p̂=(x₁+x₂)/(n₁+n₂), then z=(p̂₁−p̂₂)/√[p̂(1−p̂)(1/n₁+1/n₂)]; reject H0 if |z| exceeds the normal critical value for the chosen α.

What is testing of hypothesis? Explain the procedure followed in testing of significance difference between two population proportion for large sample.

Testing of hypothesis is a statistical procedure for deciding, from sample data, whether a claim (hypothesis) about a population parameter should be rejected or not. The claim to be tested is the null hypothesis H0; it is compared with an alternative H1 using a test statistic and a level of significance α.

Test for the difference of two population proportions (large samples)

Used to test whether proportions p₁ and p₂ in two populations are equal, from samples of sizes n₁, n₂ with x₁, x₂ successes. Samples must be large (n·p ≥ 5 and n·q ≥ 5 in each).

  1. Hypotheses. H0: p₁ = p₂. H1: p₁ ≠ p₂ (two-tailed), or p₁ > p₂ / p₁ < p₂ (one-tailed).
  2. Level of significance α (5% or 1%).
  3. Sample proportions and pooled proportion.
p̂₁ = x₁/n₁,   p̂₂ = x₂/n₂
p̂  = (x₁ + x₂)/(n₁ + n₂),   q̂ = 1 − p̂
  1. Test statistic.
z = (p̂₁ − p̂₂) / √( p̂ q̂ (1/n₁ + 1/n₂) )
  1. Critical value from the normal table: ±1.96 (two-tail, 5%), 1.645 (one-tail, 5%), ±2.576 (two-tail, 1%), 2.326 (one-tail, 1%).
  2. Decision. If the calculated z falls in the rejection region (|z| > critical value for two-tail), reject H0; otherwise do not reject H0.
  3. Conclusion in words: the two population proportions differ significantly, or they do not.

✓ Answer checked by Claude Sonnet 5.5

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