Question

What is the critical value of F for a one-tailed test at the α =.10 level, when there is a sample size of 8 for the sample with the smaller variance and a sample size of 11 for the sample with the larger sample variance?

a. 3.64
b. 2.70
c. 2.41
d. 1.65

Answer 1

b. 2.70

Explanation:

Data given and notation

`n_1 = 11` represent the sampe size for sample 1

`n_2 =8` represent the sample size for sample 2

`s_1` represent the sample deviation for sample 1

`s^2_1` represent the sample variance for sample 1

`s_2` represent the sample deviation for sample 2

`s^2_2` represent the sample variance for sample 2

`\alpha=0.1` represent the significance level provided

Confidence =0.90 or 90%

F test is a statistical test that uses a F Statistic to compare two population variances, with the sample deviations s1 and s2. The F statistic is always positive number since the variance it's always higher than 0. The statistic is given by:

`F=(s^2_1)/(s^2_2)`

System of hypothesis

We want to test for example if the variation for group 1 it's higher than the variation for group 2, so the system of hypothesis are:

H0:
`\sigma^2_1 \leq \sigma^2_2`

H1:
`\sigma^2_1 >\sigma^2_2`

Calculate the statistic

Now we can calculate the statistic like this:

`F=(s^2_1)/(s^2_2)=F_(calc)`

Calculate the critical value

Now we can calculate the critical value but first we need to calculate the degrees of freedom for the statistic. For the numerator we have
`n_1 -1 =11-1=10` and for the denominator we have
`n_2 -1 =8-1=7` and the F statistic have 10 degrees of freedom for the numerator and 7 for the denominator. And the critical value would be:

`F_(crit)=2.705`

And we can find it with the following excel code: "=F.INV(0.9,10,7)"

b. 2.70