Do the light bulbs fall short of 1000 hours?

Question

A manufacturer rates its bulbs at a mean life of 1000 hours. The production standard deviation is known to be $\sigma = 40$ hours. A consumer lab burns 100 bulbs to failure and records a mean life of 989 hours.

Test at the 5% level (two-sided) whether the true mean life differs from the 1000-hour rating, and report the z-statistic.

Accepted Answer

Known \sigma, so use a z-test. Hypotheses: H0: \mu = 1000 versus H1: \mu 
eq 1000. Standard error: 	ext{SE} = \frac{40}{\sqrt{100}} = \frac{40}{10} = 4 	ext{ hours}. Test statistic: z = \frac{989 - 1000}{4} = \frac{-11}{4} = -2.75. Since -2.75 = 2.75 1.96, reject H0. The two-sided p-value is 2\,[1 - \Phi(2.75)] = 2(0.00298) \approx 0.006. Conclusion: there is convincing evidence the bulbs fall short of their rated life, by about 11 hours on average. Takeaway. Large samples shrink the standard error (n = 100 gives SE = 4), so even a modest 11-hour shortfall lands at z = -2.75 and is clearly…

Do the light bulbs fall short of 1000 hours?

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