# Homework Help: Testing of population variance

1. Nov 30, 2017

### WWGD

If I may, hope I don't throw of this discussion with this question: how does one determine the distribution of statistics under different conditions, e.g., with this case: how do we know that when the pop. variance is known, the sample distribution satisfies the $\chi^2$ used?

2. Nov 30, 2017

### Ray Vickson

Those are theorems that are proved in more "mathematical" courses. When tailoring material to beginners, it is often necessary to just present statements like that as facts, because their proofs will often involve much more complicated and advanced mathematics than would be appropriate in a first course or an "applied" course. Also, you will typically be given access to tables of values that you can look at when solving specific problems. However, there are also lots of on-line resources that you can use if your tables do not include what you need. Most spreadsheets will have statistical packages that allow by-passing tables altogether and just calling on the spreadsheet to do the computation. Some good hand-held calculators will even have some of the standard statistical distributions included in their capabilities.

All that being said, I am going to engage in a bit of a discussion; it will not be a how-to post for solving a particular problem, but will instead be a discussion on how to structure such problems appropriately, so that you can figure out for yourself what type of analysis to do. It will be a bit long, and I will write it in the context of your current problem, but the same general notions apply to other types of problems as well. I think I can write the following without violating PF policy, because I am not going to tell you exactly how to solve your problem---only how to think about it.

OK, so you have a current system giving a true variance of $v_0 = 7.2^2$. You do some sampling from a new system and get a sample variance of $v = 3.5^2.$ Obviously, $v \neq v_0$, but are the differences "significant"? In particular, we really want to know if the new system has smaller variance than the old, and the single sample point we have suggests that it does. However, in any particular experiment you are bound to get sample variances different from $v_0$, just by chance; in fact, about half the time you will get larger values and about half the time smaller values. Therefore, actually getting a smaller value is really not a surprise. What may be a surprise is getting a value so much smaller than $v_0$. How likely is it that---if the true variance is $v_0$--- you can get a value as small as your observed $v?$ Are the chances 25%? Is the chance 1 in a million?

In a typical hypothesis test we pick a value such as 5% and ask whether our observed $v$ has a less than 5% chance of being seen when $v_0$ is really true. That is, if our observation has a chance of more than 5% we declare that we are not surprised enough to declare that the null hypothesis is false. Of course, it may be true or it may be false---we just do not have strong enough evidence. You should also realize that there is nothing special or magical about the value 5%; other values can be used, and often are. (Alternatively, we can try to give a so-called "p-value", which is the actual probability of an observation as small as the one we observed.)

So, to answer your original question, you need to figure out how likely it would be to see a value as small as $v = 3.5^2$ when the true variance is $v_0 = 7.2^2.$ If $v_0$ were true, the calculated value of $24\: v$ would have probability distribution $v_0 \; \chi^2(24)$, so that $24\: v/v_0$ would have distribution $\chi^2(24).$ Since we have available a table of percentiles for the distribution $\chi^2(24)$ we can see how likely it would be to get a value as small or smaller than the one we actually see.

I have tried to keep this discussion as free as possible of formulas. The aim is to understand intuitively what is happening and how to look at a problem. Once that has been established, then---and only then---can we turn to statistical formulas, because at that point we will have a firm grasp of what formulas to use, and how to use them. Just trying to use some textbook or on-line formulas without such understanding is a really, really bad policy.

Last edited: Nov 30, 2017
3. Nov 30, 2017

### Ray Vickson

Sorry: I did not notice that my response above was to you rather than the OP. Most of it was intended for the OP, not for you as such.

4. Dec 1, 2017

### WWGD

No problem, Ray.