t interval for a precisely measured rod

Question

An engineer measures a rod's length n = 6 times, getting a sample mean of \bar{x} = 25.0 cm and a sample standard deviation of s = 0.4 cm. The population standard deviation is unknown. Construct a 95% confidence interval for the true mean length.

Accepted Answer

Setup. With \sigma unknown and a very small sample, use the t interval with 	ext{df} = n - 1 = 5. For 95% confidence, t^{} = 2.571. Compute. The standard error is \frac{0.4}{\sqrt{6}} = \frac{0.4}{2.449} = 0.1633. Margin of error = 2.571 	imes 0.1633 = 0.42, so 25.0 \pm 0.42 = [24.58,\ 25.42]. Interpret. With only 6 measurements the critical value t^{} = 2.571 is well above 1.96, reflecting real uncertainty about the spread from so few points. Takeaway. The smaller the sample, the more the t interval widens relative to z; at 	ext{df} = 5 the gap is already sizable.

t interval for a precisely measured rod

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