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Procedure: test of single proportion

Short answer

Set H0:P=P0 and H1 two- or one-tailed; choose alpha; compute z=(p-hat-P0)/sqrt(P0Q0/n), p-hat=x/n; compare with normal critical values (±1.96 or 1.645 at 5%, ±2.576 or 2.326 at 1%); reject H0 if z lies in the rejection region and state the conclusion.

Write the procedure of testing of Hypothesis for single proportion.

The test of a single proportion checks whether the population proportion P equals a specified value P₀, using a large sample of size n with x items having the attribute.

Procedure

  1. Hypotheses. H0: P = P₀ H1: P ≠ P₀ (two-tailed), or P > P₀ (right-tailed), or P < P₀ (left-tailed)
  2. Level of significance α (5% or 1%).
  3. Test statistic. Sample proportion p̂ = x/n, Q₀ = 1 − P₀:
z = (p̂ − P₀) / √(P₀Q₀/n)

The normal approximation is valid when nP₀ ≥ 5 and nQ₀ ≥ 5. 4. Critical value from the standard normal table:

α       two-tailed   one-tailed
0.05    ±1.96        ±1.645
0.01    ±2.576       ±2.326
  1. Decision. Reject H0 if |z| > critical value (two-tailed), z > critical value (right-tailed) or z < −critical value (left-tailed). Otherwise do not reject H0.
  2. Conclusion. Express the decision in the words of the problem, for example "the proportion of defectives is significantly higher than 5%".

✓ Answer checked by Claude Sonnet 5.5

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