Pooling two samples of different sizes

Question

Two labs each estimate the same population mean using sample averages. Lab 1 used 100 observations; Lab 2 used 300 observations. Both draw from the same population, so a single observation has the same variance $\sigma^2$ everywhere, and a sample average of $n$ points has variance $\sigma^2/n$ .

What weight on Lab 1's average minimizes the variance of the pooled estimate?

Accepted Answer

The idea. Both sample averages are unbiased for the true mean, so any blend is unbiased. Minimize variance by weighting inversely to each average's variance. Find the variances. Lab 1's average has variance \sigma^2/100; Lab 2's has variance \sigma^2/300. Lab 2 is more precise because it used more data. Minimize. Weighting inversely to variance means weighting proportionally to sample size, since variance is proportional to 1/n: w1 = \frac{\sigma^2/300}{\sigma^2/100 + \sigma^2/300} = \frac{1/300}{4/300} = \frac{1}{4} = 0.25. Equivalently, w1 = \frac{n1}{n1 + n2} = \frac{100}{400} = 0.25. The …

Pooling two samples of different sizes

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