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.
If we knew the real distribution average, we could construct a sum of squares of independent identically distributed standard normal variables "\u03be_i" similar to "S^2". It would've been "\u03c7^2_n -distributed: \\frac{(n\u22121)S^2}{\u03c3^2}=\u2211_i^n(\\frac{\u03bei\u2212\u03bc}{\u03c3})^2\u223c\u03c7^2_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.
"\u03be\u00af=\\frac{\u2211_1^n\u03bei}{n}"
But this takes away one degree of freedom (if you know the sample mean, then only ξi from 1 to n−1 can take arbitrary values, but the nth has to be "\u03be_n=\u03be\u2212\u2211_1^n\u22121\u03be_i )."
So our real S2 loses one degree of freedom "\\frac{(n\u22121)S^2}{\u03c3^2}=\u2211_1^n(\\frac{\u03bei\u2212\u03be}{\u03c3})^2\u223c\u03c7^2_{n\u22121}"
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