Question
A furniture maker has completed 25 deliveries, averaging 40 days from order to arrival. For this exercise, independent delivery times follow a normal distribution with unknown mean \(\mu\) and known standard deviation 8 days. The 25 orders are a random sample, and the next customer's order follows the same distribution independently.
The manager has calculated an interval for the long-run average delivery time and wants to give that same interval to an individual customer. Construct a two-sided 95% confidence interval for \(\mu\) and a two-sided 95% prediction interval for one new delivery. Use \(z_{0.975}=1.96\). Why are the intervals different, and what happens to their widths if the historical sample grows indefinitely?
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Hint 1
The sample mean has variance \(\sigma^2/n\). One future delivery has its own variation in addition to uncertainty in the estimated mean.
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Solution
Let \(\bar T=40\), \(n=25\) and \(\sigma=8\) days. Under the stipulated normal model, \(\bar T\sim N(\mu,\sigma^2/n)\). A 95% confidence interval for the fixed population mean is
\(\bar T\pm1.96\,\sigma/\sqrt n=40\pm1.96(8/5)=40\pm3.136\),
or \(\boxed{[36.864,43.136]}\) days. This interval's repeated-sample coverage concerns \(\mu\), not a particular order.
For the next independent order \(T_{\rm new}\), the difference \(T_{\rm new}-\bar T\) has mean zero and variance \(\sigma^2+\sigma^2/n=64(1+1/25)\), since the new delivery time is independent of the historical sample. Normality therefore gives the exact (under this model) 95% prediction interval
\(\bar T\pm1.96\,\sigma\sqrt{1+1/n}=40\pm1.96(8)\sqrt{26/25}\approx40\pm15.9905\),
or \(\boxed{[24.0095,55.9905]}\) days. As \(n\to\infty\), the mean CI half-width \(1.96\sigma/\sqrt n\) tends to zero, but the prediction half-width tends to \(1.96\sigma=15.68\) days. Measuring the average more precisely cannot remove the genuine randomness of one customer's delivery time. The intervals are model-based, not a contractual guarantee that 95% of future orders fall in one fixed quoted range.
Assumptions
The historical 25 durations and the next duration are independent observations from a normal model with common fixed mean \(\mu\) and known \(\sigma=8\) days.
The confidence level and predictive coverage are frequentist over repeated independent samples; \(z_{0.975}=1.96\) is prescribed.
Final Answer
95% mean CI: 36.864–43.136 days. 95% individual prediction interval: approximately 24.009–55.991 days. The prediction interval retains a limiting half-width of 15.68 days, while the mean CI narrows to zero.