t interval for mean chemical concentration

Question

A lab measures the concentration of a compound in $n = 8$ water samples. The sample mean is $\bar{x} = 40$ mg/L with sample standard deviation $s = 6$ mg/L. The population standard deviation is unknown.

Construct a 95% confidence interval for the true mean concentration.

Accepted Answer

Setup. With \sigma unknown and a small sample, use the t interval \bar{x} \pm t^{}\,s/\sqrt{n} with 	ext{df} = n - 1 = 7. For 95% confidence, t^{} = 2.365. Compute. The standard error is \frac{6}{\sqrt{8}} = \frac{6}{2.828} = 2.121. Margin of error = 2.365 	imes 2.121 = 5.02, so 40 \pm 5.02 = [34.98,\ 45.02]. Interpret. With only 8 observations the degrees of freedom are low, so t^{} is noticeably above 1.96 and the interval is wider than a z interval would be. Takeaway. Fewer data points push t^{} higher and widen the interval, which is exactly the honesty you want when information is scar…

t interval for mean chemical concentration

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