Solve the problem.
-When performing a hypothesis test for the ratio of two population variances, the upper critical $\mathrm { F }$ value is denoted $\mathrm { F } _ { \mathrm { R } }$. The lower critical $\mathrm { F }$ value, $\mathrm { F } _ { \mathrm { L } }$, can be found as follows: interchange the degrees of freedom, and then take the reciprocal of the resulting $\mathrm { F }$ value found in table $\mathrm { A } - 5$. $\mathrm { F } _ { \mathrm { R } }$ can be denoted $\mathrm { F } _ { \alpha / 2 }$ and $\mathrm { F } _ { \mathrm { L } }$ can be denoted $\mathrm { F } _ { 1 } - \alpha / 2$.
Find the critical values $F _ { L }$ and $F _ { R }$ for a two-tailed hypothesis test based on the following values:
$$\mathrm { n } _ { 1 } = 10 , \mathrm { n } _ { 2 } = 16 , \alpha = 0.05$$
A) $3.1227,3.7743$
B) $0.2653,3.7743$
C) $0.2653,3.1227$
D) $0.3202,3.1227$

Question

Quizplus · Accepted Answer

The degrees of freedom for the two samples are $n_1 - 1 = 9$ and $n_2 - 1 = 15$. For a two-tailed test with $\alpha = 0.05$, the upper critical value $F_R$ is found using the degrees of freedom $9$ and $15$ directly from the F-distribution table. The lower critical value $F_L$ is found by interchanging the degrees of freedom (using $15$ and $9$) and taking the reciprocal of the corresponding F value. The correct values are $F_L = 0.2653$ and $F_R = 3.1227$, indicating the reciprocal relationship and the adjustment for the two-tailed test's significance level.

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