95% interval for mean commute time

Question

A commuter's trip time has a known population standard deviation of $\sigma = 6$ minutes. Over a random sample of $n = 144$ trips, the mean commute is $\bar{x} = 42$ minutes.

Construct a 95% confidence interval for the true mean commute time.

Accepted Answer

Setup. With \sigma known, use \bar{x} \pm z^{}\,\sigma/\sqrt{n} with z^{} = 1.96 for 95% confidence. Compute. The standard error is \frac{6}{\sqrt{144}} = \frac{6}{12} = 0.5. Margin of error = 1.96 	imes 0.5 = 0.98, so 42 \pm 0.98 = [41.02,\ 42.98]. Interpret. The large sample (n = 144) makes the standard error small, so the estimate of the mean commute is quite precise. Takeaway. The standard error \sigma/\sqrt{n} is the engine of the interval: fix \sigma and the level, and precision improves only as \sqrt{n} grows.

95% interval for mean commute time

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