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Test of proportion: pumps needing repairs

Short answer

Use right-tailed z-test for a proportion. p̂ = 47/120 = 0.3917, p₀ = 0.30, SE = 0.04183, z = 2.191. Critical z₀.₀₅ = 1.645. Reject H₀: p=0.30. Evidence supports p>0.30, so the manufacturer’s claim is not supported.

A manufacturer of submersible pumps claims that at most 30% of the pumps require within the first 5 years of operation. If a random sample of 120 of the pumps includes 47 which required repairs within the first 5 years, test the null hypothesis p = 0.30 against the alternative hypothesis p > 0.30 at the 0.05 level of significance.

Large-sample, right-tailed test of a single proportion.

n = 120, x = 47, p̂ = 47/120 = 0.3917,  p₀ = 0.30, q₀ = 0.70
  1. H0: p = 0.30 H1: p > 0.30 (more than 30% of pumps need repair)
  2. Level of significance: α = 0.05
  3. Test statistic:
SE = √(p₀q₀/n) = √(0.30×0.70/120) = 0.04183
z  = (p̂ − p₀)/SE = (0.3917 − 0.30)/0.04183 = 2.191
  1. Critical value: z(0.05, right-tail) = 1.645. Reject H0 if z > 1.645.
  2. Decision: z = 2.191 > 1.645 (p-value = 0.014), so reject H0.

Conclusion: at the 5% level the data show that more than 30% of the pumps need repair within the first 5 years, so the manufacturer's claim is not supported.

✓ Answer checked by Claude Sonnet 5.5

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