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Critical value; water pumps conforming

Short answer

Test H0: p=0.95 vs H1: p<0.95. Sample p̂=228/250=0.912, SE=√(0.95×0.05/250)=0.013784, z=−2.757. Critical value at 1% is −2.326; reject H0, manager’s doubt is justified.

Define critical value. A manufacturer claimed that at least 95% of the water pumps supplied to the ABC Company conformed to specification. However, the product manager at ABC Company wasn't satisfied with the claim of the manufacturer hence to test the claim, the manager examined a sample of 250 water pumps supplied last month and found that 228 water pumps as per the specification. Can you conclude that the production manager is right to doubt on the claim of the manufactures (α = 0.01)

Critical value: the value of the test statistic (read from the z, t, χ² or F table for the chosen α and degrees of freedom) that separates the rejection region from the acceptance region. If the calculated statistic falls beyond it, H0 is rejected. For example, ±1.96 for a two-tailed z-test at α = 0.05.

Test of the claim

The claim is p ≥ 0.95, so the doubt is that p < 0.95 (left-tailed test, large sample).

n = 250, x = 228, p̂ = 228/250 = 0.912,  p₀ = 0.95, q₀ = 0.05
  1. H0: p = 0.95 (the claim holds, at least 95% conform) H1: p < 0.95 (fewer than 95% conform; the manager's doubt)
  2. Level of significance: α = 0.01
  3. Test statistic:
SE = √(p₀q₀/n) = √(0.95×0.05/250) = √0.00019 = 0.013784
z  = (p̂ − p₀)/SE = (0.912 − 0.95)/0.013784 = -2.757
  1. Critical value: z(0.01, left-tail) = −2.326. Reject H0 if z < −2.326.
  2. Decision: z = -2.757 < −2.326 (p-value = 0.0029), so reject H0.

Conclusion: at the 1% level the sample does not support the manufacturer's claim. The production manager is right to doubt that at least 95% of the water pumps conform to the specification.

✓ Answer checked by Claude Sonnet 5.5

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