Question

Two sample of size 9 and 12 are drawn from two normally distributed populations having varriances 20 and 8,determine whether the first sample has a significantly larger varriance than the second sample at significance levels of; (a) 0.005
(b)0.001

Answer 1

The F-hypothesis test is defined for this question as:

$H_0: σ_1^2 = σ_2^2$

$H_1: σ_1^2 > σ_2^2$

Given that,

$n_1=9,n_2=12$

$s_1^2 = 20,s_2^2=8$

F test statistic, $=s_1^2 / s_2^2$

$=20/8=2.5$

a) For $\alpha=0.005$ and degree of freedoms $(8,11)$

Critical region will be $F>F(\alpha,n_1-1,n_2-1)$

$F(\alpha,n_1-1,n_2-1)=F(0.005,8,11)=5.6821$

So, the critical region is $F>5.6821$

Since the test statistics does not fall in the critical region, null hypothesis cannot be rejected. So, there is not enough evidence to show that first variance is larger than the second variance.

b) For $\alpha=0.001$ and degree of freedoms $(8,11)$

Critical region is $F>8.354$

Since the test statistics does not fall in the critical region, null hypothesis cannot be rejected. So, there is not enough evidence to show that first variance is larger than the second variance.