Inference for Two Variances: An Example of a Confidence Interval and a Hypothesis Test

I work through an example of a confidence interval and a hypothesis test for the ratio of population variances, using F procedures that are based on the assumption of normally distributed populations.
The example in this video involves tail lengths of male and female lizards. The summary statistics and distributions of the tail lengths are from Table 1 and Figure 2 in:
Qu et al. (2011). Sexual dimorphism and female reproduction in two
sympatric toad-headed lizards, Phrynocephalus frontalis
and P. versicolor (Agamidae). Animal Biology. 61:139-151.

[@nourelislam8565]

Awesome video ........but I have a question !!!
Is the difference in the sample size between the samples which is 22 for male and 44 for females affect in the value of the variance between them??

[@jbstatistics]

There is no fundamental problem with having different sample sizes (the procedure works fine with different sample sizes). There is often an advantage to having equal sample sizes in statistics, in that it results in greater efficiency of estimators of a difference, but it's not necessary. It's much better to have sample sizes of 22 and 44, say, than 22 and 22.

[@nourelislam8565]

@@jbstatistics thanks alot

[@ab12346781]

This material is really good. Thank you very much!!! A suggestion is that it should be indexed.

[@jbstatistics]

Thanks for the compliment. The complete set of videos is organized in better fashion on jbstatistics.com.

[@quanesshatheseventeenth8617]

why is the critical value F here instead of X^2?

[@jbstatistics]

It's just the way the math works out. The ratio of two independent sample variances from normally distributed populations with equal variances has an F distribution. (Equivalently, the ratio of two independent chi-square random variables divided by their respective degrees of freedom has an F distribution.) Full mathematical details of this are typically covered in a first course in mathematical statistics.

[@quanesshatheseventeenth8617]

@@jbstatistics Got it. Also is there an independence condition like “n < 10N” or “Expected successes and failures being at least 5” for the confidence interval for a ratio of variances or the confidence interval of variances?

[@jbstatistics]

@@quanesshatheseventeenth8617 We need independent random samples from normally distributed populations. This procedure is extremely sensitive to the normality assumption, even for larger sample sizes. So it's always a bit of a sketchy procedure.

[@starboyza4716]

Do we always use 95%

[@quanesshatheseventeenth8617]

Not necessarily, its just the safest one and most commonly used one.