I'm not sure how to answer the question below. I have gone through worked examples and I have the answer for this question as well. However, the answer doesn't really provide any working.
There are similar questions to this that are basically structured in the same way: 'what is the probability that a sample variance this high/low would be found if the population variance is x'. I'm assuming the same process is used to solve them.
Some context: I am an undergraduate student and I was working through my book when I came across this question. Relating to sample variance, I know that:
Given a random sample of 'n' observations from a normally distributed population whose population variance is $\sigma^2$ and whose resulting sample variance is $s^2$, it can be shown that: $$\chi^{2} = \frac{(n-1)s^2}{\sigma^2}$$ has a chi-square distribution with n-1 degrees of freedom.
I know for the below question I may have to find values for the cumulative distribution of $\chi^{2}$, which could be either the upper or lower tail of the distribution: P($\chi^{2}$<K) = 0.05 for example or P($\chi^{2}$>K) = 0.05 [where K can be any critical value based on the (n-1) degrees of freedom, for example: 3.94 for the lower distribution, when (n-1)=10].
I'm not sure if I have to use an upper or lower tail for the below question and what value to find from the above expression.
It would be great if you could explain the process behind solving this question:
Q. A manufacturer has been purchasing raw materials from a supplier whose consignments have a variance of 15.4 (in squared pounds) in impurity levels. A rival supplier claims that she can supply consignments of this raw material with the same mean impurity level but with lower variance. For a random sample of 25 consignments from the second supplier, the variance in impurity levels was found to be 12.2. What is the probability of observing a value this low or lower for the sample variance if, in fact, the true population variance is 15.4? Assume that the population distribution is normal.
[if this helps: Supplier 1: variance: 15.4; Supplier 2: n=25, variance: 12.2; true population variance: 15.4]
Also, I apologise in advance if this isn't the right stack exchange to ask this question. Let me know if it's more appropriate in another stack exchange (as I posted it here since it part of my statistics course in economics).
Any help you could provide would be much appreciated.
Thanks a lot!