t interval for a mean with a small sample

Question

A trader records the returns on n = 10 independent trades. The sample mean is \bar{x} = 2.5\% and the sample standard deviation is s = 1.8\%. The population standard deviation is unknown. Construct a 95% confidence interval for the true mean return.

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Setup. Because \sigma is unknown and estimated by s, and the sample is small, use the t distribution: \bar{x} \pm t^{}\,\frac{s}{\sqrt{n}}, \qquad 	ext{df} = n - 1 = 9. For 95% confidence with 9 degrees of freedom, t^{} = 2.262. Compute. The standard error is \frac{s}{\sqrt{n}} = \frac{1.8}{\sqrt{10}} = \frac{1.8}{3.162} = 0.569. Margin of error = 2.262 	imes 0.569 = 1.29, so 2.5 \pm 1.29 = [1.21,\ 3.79]\ 	ext{percent}. Interpret. The t value 2.262 is larger than the z value 1.96. That extra width is the price of not knowing \sigma: it accounts for the added uncertainty from estimating the…

t interval for a mean with a small sample

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