# F-distribution vs chi-square

Hi everyone, I have some trouble understanding the following question from Investopedia:

​Which of the following statements with respect to test statistics are correct? I. The test statistic is the figure that will be used to decide whether or not to accept the null hypothesis. II. The chi-square distribution is the test used to determine if the variances of two populations are the same. III. The F-distribution tests whether or not two characteristics from a population are independent of each other. IV. A normal distribution may only be used if the population standard deviation is known, the exception being if the sample size drawn from a normal population is large. (a) I and IV only (b) I, II and III © I, II, III and IV

​The correct answer is C), so all statements are correct. But I am almost certain that II and III are not correct (or should be switched). It was my understanding that the chi-square distribution is used to test whether two characteristics from a population are independent of each other and the F-Test is used to determine if the variances of two populations are the same. Alternatively, we can use the Chi-Square distribution to test if the variance of a population is equal to a certain value.

​Any ideas? Thanks,

​Tartaglia

In fact, I just found this in the Reading 11 (Hypothesis Testing), p 636 of the CFA curriculum:

​"In tests concerning the variance of a single normally distributed population, we make use of a chi-square test statistic, denoted χ2"

“Tests concerning the difference between the variances of two populations make use of the F-distribution”

Firstly, while this question may be good for generally testing your knowledge, CFA Institute specifically mentions that these types of answer choices will not appear on the actual exam ! See first paragraph under bold heading “Answer Choices” of the document in the link below.

https://www.cfainstitute.org/programs/cfaprogram/exams/Documents/level_I_questions_formatting_conventions.pdf

The curriculum uses the F test when testing the equality/inequality of two population variances and the chi-squared test for a test concerning a single population variance (all populations are assumed to be normally distributed for these tests)

So I personally wouldn’t bother reading too much into these answers. But for the record, I think A) is correct.

II is definitely false.

III is so vague as to be useless.

I and IV are OK, though IV is not well-written.

Thank you both for your quick responses, I agree with you two, I would have also picked I and IV.

​Thank you also for the hint with the question types, that is very good to know.

My pleasure.

I wouldn’t really say that I is okay. It’s a well known guideline to never accept the null hypothesis; you either reject or fail to reject the null. The tests we conduct can’t ever prove the null true (leading to acceptance).

It’s perfectly acceptable if they change the wording from “accept” to “reject”.

Good point.

All-in-all, a poorly written question.

I agree. I’ve heard (no real personal experience) that Investopedia isn’t very accurate with the CFA exam material (and some non-CFA topics), which is a shame to hear about a free resource.

IV also doesn’t look very nice. I may be reading it wrong, but a large sample would allow us to use a normal distribution (irrespective of the true distribution’s shape) based on the central limit theorem (i.e. the sample only needs to be large, not also from a normal distribution).

Also, III somewhat looks like it should be a Chi-Squared test to determine the independence of two qualitative variables (for example, determining if coffee drinking status (Yes/No) is infuenced by gender (Male/Female)). You could test this using a Chi-squared test for heterogeneity/independence. Granted, this is beyond the curriculum, but maybe it’ll give some context to what they might mean by two “characteristics” being independent of one another. A finance context: does the bond issue (government vs non-government) influence the default status on an issue (defaulted vs not)?

Yes: Investopedia has some glaring errors.

I agree about the quality of questions on Investopedia. Regarding the “acceptance of the null”: I worked my may through the entire bank of Quantitative Questions in their CFA section, and I think it is safe to say, that you won’t be able to reject the hypothesis that they never ‘fail to reject the null’ but rather always ‘accept the null’, so by now I have come to terms with their wording (it used to drive me nuts). But in any case, they still have a large question bank on all sections so I figure it cannot hurt.

I may be misunderstanding your statement, but are you saying that they are teaching you to “accept the null” instead of saying “fail to reject the null”?

Yes (sorry about the lack of clarity in my statement). It was pounded in my head in countless statistics and econometrics classes to ‘fail to reject the null’, so as I said, I just started to ignore the ‘accepting the null’ in their questions. But if this is the primary source of preparation for anyone, it will take a while to get that out of their heads. Here is a sample of questions (which is obviously not representative, you have to take my word for it):

​Which of the following statements least accurately describes the interpretation of a significance level? (a) As the level of significance increases, we increase the possibility of rejecting the null when in fact it is true. (b) The higher the level of significance, the lower will be the confidence interval. © A low level of significance makes it harder to accept the null hypothesis. (d) Hypothesis test results that were conducted with a very low level of significance would not reveal much useful information.

​A CEO of a large corporation wishes to examine the relationship between her company’s sales and GDP. Looking at the “monthly” data over the last five years, she determines that the correlation coefficient between company sales and the economy is 0.53. With a level of significance of 5%, is it possible that there really is no correlation between company sales and GDP? A) Since t-calc of 4.76 falls outside the critical t-values, the statement that company sales are not correlated with GDP should be accepted. B) Since t-calc of 1.08 falls outside the critical t-values, the statement that company sales are not correlated with GDP should be rejected. C) Since t-calc of 4.76 falls outside the critical t-values, the statement that company sales are not correlated with GDP should be rejected. D) Since t-calc of 1.08 falls within the critical t-values, the statement that company sales are not correlated with GDP should be accepted.

​Which of the following statements least accurately describes the interpretation of a significance level? (a) As the level of significance increases, we increase the possibility of rejecting the null when in fact it is true. (b) The higher the level of significance, the lower will be the confidence interval. © A low level of significance makes it harder to accept the null hypothesis. (d) Hypothesis test results that were conducted with a very low level of significance would not reveal much useful information.

I see what you mean. Luckily, your prior education taught you the correct phrase of “fail to reject the null.” People who tell you there is no difference between the statements either don’t really know what it means or are being very careless.

Agreed. Thanks again for pointing that out.