95% interval for mean battery life with known sigma

Question

A battery manufacturer knows from production history that lifetime has a population standard deviation of $\sigma = 40$ hours. A random sample of $n = 64$ batteries has a mean lifetime of $\bar{x} = 520$ hours.

Construct a 95% confidence interval for the true mean battery life.

Accepted Answer

Setup. With \sigma known, use the z interval \bar{x} \pm z^{}\,\sigma/\sqrt{n} with z^{} = 1.96 for 95% confidence. Compute. The standard error is \frac{40}{\sqrt{64}} = \frac{40}{8} = 5. Margin of error = 1.96 	imes 5 = 9.8, so the interval is 520 \pm 9.8 = [510.2,\ 529.8]. Interpret. We are 95% confident, in the sense of the long-run procedure, that the true mean lifetime lies in this range. Takeaway. Larger samples shrink the standard error and tighten the interval; here \sqrt{64} = 8 divides the raw spread down to a small margin.

95% interval for mean battery life with known sigma

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