Question

Two random samples were drawn from two normal populations
and the values are
A 66 67 75 76 82 84 88 90 92
B 64 66 74 78 82 85 87 92 93 95 97
Test whether the two populations have the same variance at the 5%
level of significance (F=3.36) at 5% level for v1=10 and v2 = 8.

Answer 1

$$\bar{x}_1=\dfrac{1}{9}(66+ 67+ 75 +76 +82+ 84 +88$$

$$+90+ 92)=80$$

$$\bar{x}_2=\dfrac{11}{9}(64+ 66 +74+ 78 +82 +85 +87$$

$$+ 92+ 93+ 95+ 97)=83$$

$$s_1^2=\dfrac{1}{9-1}((66-80)^2+(67-80)^2+(75-80)^2$$

$$+(76-80)^2 +(82-80)^2+(84-80)^2$$

$$+(88-80)^2 +(90-80)^2 +(92-80)^2$$

$$=\dfrac{367}{4}=91.75$$

$$s_2^2=\dfrac{1}{11-1}((64-83)^2+(66-83)^2+(74-83)^2$$

$$+(78-83)^2 +(82-83)^2+(85-83)^2$$

$$+(87-83)^2 +(92-83)^2 +(93-83)^2$$

$$+(95-83)^2+(97-83)^2$$

$$=\dfrac{649}{5}=129.8$$

$$F=\dfrac{s_2^2}{s_1^2}=\dfrac{129.8}{91.75}\approx1.4147$$

For $\nu_1=8$ and $\nu_2=10, F_{0.05}=3.36$

The calculated value of $F$ is less than the tabulated value. So the null hypothesis is accepted.

Hence it may be concluded that two populations have the same variance.