Question

Two varieties of wheat are being tested in a developing country. Twelve test plots are given identical preparatory treatment. Six plots are sown with Variety 1 and the other six plots with Variety 2 in an experiment in which the crop scientist hope to determine whether there is significant difference between yields. Consider the following sample statistics: . What is the standard deviation of ? Assume equal variance.

Answer 1

If we knew the real distribution average, we could construct a sum of squares of independent identically distributed standard normal variables $ξ_i$ similar to $S^2$. It would've been $χ^2_n -distributed: \frac{(n−1)S^2}{σ^2}=∑_i^n(\frac{ξi−μ}{σ})^2∼χ^2_n$ with n degrees of freedom.

However, we need to know the real distribution mean μ and distribution variance σ² to achieve the desired sum, and they are not available in real-life small sample situations.

As we don't know the real distribution mean μ, you have to approximate it with sample average.

$ξ¯=\frac{∑_1^nξi}{n}$

But this takes away one degree of freedom (if you know the sample mean, then only ξ_i from 1 to n−1 can take arbitrary values, but the nth has to be $ξ_n=ξ−∑_1^n−1ξ_i ).$

So our real S² loses one degree of freedom $\frac{(n−1)S^2}{σ^2}=∑_1^n(\frac{ξi−ξ}{σ})^2∼χ^2_{n−1}$