Testing null hypothesis \({H_0}:\sigma _1^2 = \sigma _2^2\) versus alternative hypothesis\({H_a}\), under the assumption that the two populations are normal and independent, can be performed using test statistic value
\(f = \frac{{s_1^2}}{{s_2^2}}\)
Depending on alternative hypothesis \({H_a}\) the \(P\)value is corresponding area under the \({F_{m - 1,n - 1}}\)curve.
The hypotheses of interest are \({H_0}:\sigma _1^2 = \sigma _2^2\)versus\({H_a}:\sigma _1^2 \ne \sigma _2^2\). The missing values to compute the \(f\) value are the sample standard deviations \({s_1}\)and \[{s_2}\]
The Sample Variance \[{s_2}\] is
\({s^2} = \frac{1}{{n - 1}} \cdot {S_{xx}}\)
where
\({S_{xx}} = \sum {{{\left( {{x_i} - \bar x} \right)}^2}} = \sum {x_i^2} - \frac{1}{n} \cdot {\left( {\sum {{x_i}} } \right)^2}\)
The Sample Standard Deviation\(s\)is
\(s = \sqrt {{s^2}} = \sqrt {\frac{1}{{n - 1}} \cdot {S_{xx}}} \)
Thus, since\(m = 10\)and\[n = 5,\]the sample standard deviations are
\({s_1}{\rm{ }} = \sqrt {\frac{1}{{10 - 1}} \cdot \left[ {\left( {{{29}^2} + {{34}^2} + \ldots + {{27}^2}} \right) - \frac{1}{{10}} \cdot {{(29 + 34 + \ldots + 27)}^2}} \right]} \)
\( = 2.75\)
and
\({s_1}{\rm{ }} = \sqrt {\frac{1}{{5 - 1}} \cdot \left[ {\left( {{{18}^2} + {{15}^2} + \ldots + {{12}^2}} \right) - \frac{1}{5} \cdot {{(18 + 15 + \ldots + 12)}^2}} \right]} \)
\( = 4.44\)
