Question

7. Two random samples drawn from two normal populations are
Sample I 20 16 26 27 23 22 18 24 25 19 - -
Sample II 27 33 42 35 32 34 38 28 41 43 30 37
Obtain the estimates of the variances of the population.

Answer 1

$S^{2}=\frac{1}{n-1}(\sum x^{2}-\frac{(\sum x)^{2}}{n})\\ \text{For the first sample n=10}\\ S_{1}^{2}=\frac{1}{9}(20^{2}+16^{2}+...+19^{2}-\frac{(20+16+...+19)^{2}}{10})\\ =13.33\\ \text{For the second sample n=12}\\ S_{2}^{2}=\frac{1}{11}(27^{2}+33^{2}+...+37^{2}-\frac{(27+33+...+37)^{2}}{12})\\ =28.55\\ S_{p}^{2}=\frac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{(n_{1}+n_{2}-2)}\\ S_{p}^{2}=\frac{(9)(13.33)+(11)(28.55)}{20}\\ =21.701$