Which statement about the F F -distribution is always true?

Everybody. So now that we've learned the basics of the f distribution, let's take a look at some practice problems. We're gonna get some more computation ones in just a second, but now here's a conceptual one. So we're gonna ask what statement about the f distribution is always true. Now if you've already gotten the right answer for this, go ahead and just move on. But if not, let's go ahead and talk through the options here. We've got option a, which says that it's symmetric just like the normal distribution. Option b says it's centered at zero. C says that it's always right skewed and takes only positive values. And option d says it becomes negative when the denominator variance is very small. Alright. So let's just briefly recap our distributions. If I were to draw something like the normal distribution, it would look like this, like the familiar sort of bell shaped curve that we're always so used to seeing. It has a clear sort of center because it's symmetrical, and therefore, because it's centered at zero, we have positive and negative values. We could have positive or negative z scores. Now we know our f distribution looks nothing like this. In fact, it actually looks very similar to our chi square distribution. So what happens here is we have a distribution that kind of goes off to the right like this, and instead of being centered, it actually sort of is like a right skewed. It kind of like rises to a peak and then slowly sort of tapers off as it moves towards the right. Alright? Now what happens here is this actually goes all the way off to positive infinity, and off to the left side, the smallest possible f distribution value you could possibly get is zero. So that being said, let's go ahead and take a look at a couple of our answer choices and see if we can already start eliminating some. Option a said it's symmetric just like the normal distribution. Well, clearly, we can tell here that no matter what the degrees of freedom are, this is always going to be right skewed. Right skewed meaning it has a long right tail. Alright? Now I did mention that for varying degrees of freedom, for higher degrees of freedom, this thing can start to look like it is normally distributed, but it never really quite gets there. So even though it starts to look a little bit symmetric, it actually isn't, and it's always going to be right skewed. Alright? So there therefore, option a is definitely not the right answer. Now what about option b? Says it's centered at zero. Remember, the normal distribution is centered at zero and we have positive and negative z scores. The f distribution is not like that at at all. We said that the smallest possible value you can get was zero, and there actually is no cap in terms of how large your f numbers can get. So it's definitely not centered at zero. Alright? Now what about the third option? It says that it's always right skewed and takes only positive values. Well, it's definitely a possibility given what we just talked about. It's right skewed, and it takes positive top positive values, but let's take a take a look at our fourth option here, which says that it becomes negative when the denominator variance is very small. Now what we said here is that if you have an f score like this, then we found the p value by taking the area to the right of that test statistic, and we saw how to do that on our graphing calculators. Now what this f statistic represents is really just a ratio of variances. It is s one squared over s two squared. And one of the things we mentioned here is that in an f statistic, like a hypothesis test, typically, the convention is that s one squared is always the larger variance. What happens here is that regardless of which one is on top or the bottom, variances are always gonna be positive numbers. So because variance is a number that is squared, you're always gonna have two numbers that are positive on top of each other. Basically, what that means is that you can never have f statistics that aren't negative. That's exactly why on our graphs, zero is the smallest possible f statistic you can possibly get. Alright? So, therefore, d is definitely wrong. It definitely becomes, it cannot become negative, no matter what the denominator variance is. Torian's choice is c, which is that it's always right skewed and only takes only positive values. Have you got that? Great job. So let's move on to a little bit more computational problems. I'll see you there.

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1. Introduction to Statistics53m

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Uniform Distribution
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Two Variances and F Distribution

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10. Hypothesis Testing for Two Samples

Two Variances and F Distribution

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Multiple Choice

Which statement about the $F$ -distribution is always true?

It is symmetric like the normal distribution.

It is centered at 0.

It is always right-skewed and takes only positive values.

It becomes negative when the denominator variance is very small.

Next problem

Verified step by step guidance

Understand that the F-distribution is used primarily to compare variances between two populations, often in the context of ANOVA or hypothesis testing about variances.

Identify the degrees of freedom for the numerator (df1) and the denominator (df2). These typically come from the sample sizes of the two groups: df1 = n1 - 1 and df2 = n2 - 1.

Determine the critical value or the p-value from the F-distribution table or software using the degrees of freedom and the significance level (usually denoted as $ \alpha $).

Calculate the F-statistic using the formula $ F = \frac{S_1^2}{S_2^2} $, where $ S_1^2 $ and $ S_2^2 $ are the sample variances of the two groups, ensuring that the larger variance is in the numerator to get an F value greater than or equal to 1.

Compare the calculated F-statistic to the critical value to decide whether to reject the null hypothesis that the variances are equal, or use the p-value to assess significance.