Small normal samples
Section of Estimation examples, part of Data analysis
Small normal samples
| Type | Section |
| Taught in | Estimation examples |
| Part | Data analysis |
Example 1 (The volume of a drop) A burette reads 25.36 mL, then 22.84 mL after 50 drops, then 20.36 mL after another 50.
Question. Give a 95% interval for the volume of one drop.
Solution. The two batches of 50 drops have volumes x_1=2.52 and x_2=2.48 mL. Model them as a \operatorname{N}(\mu,\sigma^2) sample of size n=2, where \mu is the expected volume of 50 drops. Then \overline x=2.50, s=0.02\sqrt2, and the standard error is s/\sqrt2=0.02. With one degree of freedom t_{1;0.975}=12.71, so the interval for \mu is
2.50\pm12.71\times0.02=(2.25,\ 2.75).
Dividing by 50, a monotone change of scale, gives a 95% interval for one drop of (0.045,\ 0.055) mL. The huge quantile 12.71 is the price of estimating a standard deviation from two numbers.
The t and chi-squared quantiles come from R:
qt(0.975, df = c(1, 2, 5, 50, 100)) # 12.71 4.30 2.57 2.01 1.98
qchisq(c(0.025, 0.975), df = 24) # 12.40 39.36The t quantiles fall quickly towards the normal 1.96 as the degrees of freedom grow. The chi-squared quantiles give the interval for a variance, as in the aspirin example of Confidence intervals.
Used in
See also
Margin of error · Welch interval · From data to an estimate · Large-sample intervals · Two samples with unequal variances · Intervals for proportions · Wilcoxon rank-sum test