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Test of proportion: 95% equipment claim

Short answer

Using a left-tailed one-proportion z test with p̂=0.91 and p0=0.95, z=-2.596 is below the critical value -1.645; reject H0. The sample does not support the claim that at least 95% conform.

A manufacturer claimed that at least 95% of the equipment, which he supplied to a factory, conformed to specifications. An examination of a sample of 200 pieces of equipment found that 18 are faulty. Test his claim at 5% level of significance.

Left-tailed test of a single proportion (large sample). Equipment conforming = 200 − 18 = 182.

n = 200, x = 182, p̂ = 182/200 = 0.91,  p₀ = 0.95, q₀ = 0.05
  1. H0: p = 0.95 (at least 95% conform; claim is true) H1: p < 0.95 (fewer than 95% conform; claim is false)
  2. Level of significance: α = 0.05
  3. Test statistic:
SE = √(p₀q₀/n) = √(0.95×0.05/200) = 0.01541
z  = (p̂ − p₀)/SE = (0.91 − 0.95)/0.01541 = -2.596
  1. Critical value: z(0.05, left-tail) = −1.645. Reject H0 if z < −1.645.
  2. Decision: z = -2.596 < −1.645 (p-value = 0.0047), so reject H0.

Conclusion: the sample does not support the claim; at the 5% level fewer than 95% of the equipment conform to specifications.

✓ Answer checked by Claude Sonnet 5.5

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