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Confidence level; bulb superiority claim

Short answer

Confidence level (1−α) is the probability an interval contains the parameter; significance level α is Type I error probability. Right-tailed z-test: SE=√(27²/40+31²/40)=6.5, z=1.385 < critical 2.326; fail to reject H0, the claim is not substantiated at 1%.

Define confidence level and significance level. A company claims that its light bulbs are superior to those of its main competitor. If a study showed that a sample of 40 of its bulbs has mean lifetime of 647 hours of continuous use with standard deviation of 27 hours, while a sample of 40 bulbs made by its main competitor had mean lifetime of 638 hours of continuous use with standard deviation of 31 hours. Does this substantiate the claim at 1% level of significance?

Confidence level and significance level

  • Confidence level (1 − α) is the probability (usually 90%, 95% or 99%) that a confidence interval, constructed by the given method, contains the true population parameter. A 95% level means that 95 out of 100 such intervals would capture the parameter.
  • Significance level (α) is the maximum probability of rejecting a true null hypothesis (Type I error), fixed before the test, commonly 0.05 or 0.01. It is the size of the rejection region. Confidence level = 1 − α.

Test of the company's claim

The company (sample 1) claims superiority, so a right-tailed z-test is used (both samples large).

  1. H0: μ₁ = μ₂ (the company's bulbs are not better) H1: μ₁ > μ₂ (the company's bulbs last longer) (right-tailed)
  2. Level of significance: α = 0.01
  3. Test statistic (large samples, z-test):
z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
SE = √(27²/40 + 31²/40) = √(18.2250 + 24.0250) = 6.5000
z  = (647 − 638) / 6.5000 = 1.385
  1. Critical value: z at α = 0.01 (right-tail) = 2.326. Reject H0 if z > 2.326.
  2. Decision: the calculated value falls in the acceptance region (p-value = 0.0831), so fail to reject H0.

Conclusion: the sample does not substantiate the claim at the 1% level; the observed 9-hour difference may be due to sampling variation.

✓ Answer checked by Claude Sonnet 5.5

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