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Bootstrap variance without a computer

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A sample is $1$ , $4$ , $7$ , $12$ . A resample draws $4$ values from it with replacement, and all resamples are counted. What is the variance of the resample mean? Answer as a fraction in lowest terms.

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Hint

Treat the sample as the population, then use the variance of a mean of independent draws.

Solution

The sample mean is $\frac{24}{4} = 6$ , and the squared deviations sum to $25 + 4 + 1 + 36 = 66$ .
As a population, the sample has variance $\frac{66}{4} = \frac{33}{2}$ .
The mean of $4$ independent draws has variance $\frac{\frac{33}{2}}{4} = \frac{33}{8}$ , so the answer is $\frac{33}{8}$ .

Answer: $\frac{33}{8}$

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