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Worked solution

Test a longer-life bulb claim using CLT

For many years the Consumer Union has found that the mean lifetime of a 100-watt light bulb is 72 days of continuous use, with standard deviation 12 days. The company markets a “new and improved” bulb yielding longer life. A sample of 55 new bulbs has average lifetime 75.2 days. Does this provide sufficient evidence of superiority at the 5% significance level?

CJC · Discussion · Question 2

Answer

Reject H0H_0 at 5%; upper-tail p≈0.0240p\approx0.0240 supports a mean lifetime above 72 days.

State the model and test direction

Let μ\mu be the new bulbs’ population mean lifetime. Test H0:μ=72H_0:\mu=72 against H1:μ>72H_1:\mu>72. As in the source, assume the new population still has SD 12 days and the 55 observations are independent and representative. Normality of individual lifetimes is not given; use Xˉ≈N(72,122/55)\bar X\approx N(72,12^2/55) under H0H_0 by CLT.

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Both numerical decision methods

z=(75.2−72)/(12/55)=1.977652930z=(75.2-72)/(12/\sqrt{55})=1.977652930. The upper-tail p-value is 0.02398393687≈0.0240≤0.050.02398393687\approx0.0240\le0.05, so reject. Equivalently, 1.9777>1.6448536271.9777>1.644853627, the 5% upper critical value. The source Z-Test → Stats entries are μ0=72\mu_0=72, σ=12\sigma=12, xˉ=75.2\bar x=75.2, n=55n=55, alternative >. There is sufficient evidence at 5% of a mean lifetime above 72 days. “Superior” here refers only to longer mean life, not every aspect of quality.

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5% upper lifetime test: standard normal curve, right rejection tail area 0.05, critical z 1.644853627 and observed z 1.977652930.
5% upper lifetime test: standard normal curve, right rejection tail area 0.05, critical z 1.644853627 and observed z 1.977652930.
5% upper lifetime test: standard normal curve, right rejection tail area 0.05, critical z 1.644853627 and observed z 1.977652930.

Common mistakes

The historical SD need not automatically carry over to a changed product. This is an assumption of the source test, and n 55 does not guarantee an accurate normal tail for every possible population.

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