# Understanding F-test to compare variances (R's var.test)

I gave run the following codes in R console:

#for built-in matrix ToothGrowth
tooth.1mg <- subset(ToothGrowth, dose==1)
var.test(len~supp,data=tooth.1mg,alternative="two.sided")


I have found the result:

F test to compare two variances
data:  len by supp

F = 2.4176, num df = 9, denom df = 9, p-value = 0.2046

alternative hypothesis: true ratio of variances is not equal to 1

95 percent confidence interval:
0.6004952 9.7332038

sample estimates:
ratio of variances
2.41759


But I do not understand what the result expresses. Is that we fail to reject null hypotheses since p-value>sig.level? If so, then I found in most cases by implying the command var.test that we fail to reject null hypotheses since p-value > sig.level. Or, if I focus on confidence interval, it doesn't include hypothesized value which concludes that we reject null hypotheses in favor of alternative hypotheses at 0.05 significance level.

## migrated from math.stackexchange.comMay 26 '13 at 1:59

This question came from our site for people studying math at any level and professionals in related fields.

If you read the documentation (type ?var.test or help("var.test") in the console) or look at the output carefully, you will see that it tests whether the ratio between the two variances is 1 (and not whether the difference is 0). The confidence interval therefore does include the hypothesized value (namely 1) and is perfectly coherent with the p value. In both cases, you fail to reject $H_0$ at the specified error level. See also a previous question on this test