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Confidence level; cables conforming claim

Short answer

Confidence level is 1-alpha, the probability an interval contains the parameter; significance level alpha is P(Type I error). For H0:p=0.95 versus H1:p<0.95, z=-2.757; critical value at 0.01 is -2.326, so reject H0. The manager is right to doubt the claim.

Define confidence level and significance level. A manufacturer claimed that at least 95% of the cables supplied to the ABC Company confirmed to specifications. However, the production 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 cables supplied last month and found that 228 cables as per the specifications. Can you conclude that the production manager is right to doubt on the claim of the manufacturer? (α = 0.01)

Confidence level: the probability (1 − α), such as 95% or 99%, that an interval estimate will contain the true population parameter if sampling is repeated many times. Significance level (α): the probability of rejecting a true null hypothesis (Type I error), such as 5% or 1%. The confidence level equals 1 − α.

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 cables conform to the specification.

✓ Answer checked by Claude Sonnet 5.5

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