Probability of Sample Variance Given Variance

Question

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated!

Here's the Problem: Suppose a considerable amount of effort is conducted to decrease the variability in a system. Following this, a random sample of size $n=40$ is taken from the new assembly line and the sample variance is:

$$S^{2}=0.188$$

Do we have a strong numerical evidence that $σ^{2}$ has been reduced below $1.0$?

Consider this Probability and give your conclusion:

$$P(S^{2} \leq 0.188 \mid σ^{2}=1.0)$$

Data that I have gotten:

$$n=40, S^{2}=0.188, σ^{2}=1.00, μ=9$$ Confidence Interval $= 9\pm 1.5$

I am not sure how to solve this problem, I was going to try and use the T-distribution some how but I cannot figure it out, so any help would be awesome!

Thank you

Answer 1

You need to conduct a hypothis test for the variance not for the mean. The confidence interval $9\pm 1.5$ is a confidence interval for the mean and will not help you to draw conclusions about the variance! Accordingly, you do not need the $t$-distribution but the $\chi^2$-distribution. Formally

In order to conclude whether $σ^2$ has been reduced below $1.0$ we will statistically test the null hypothesis $$H_0:σ^2=1.0$$ against the alernative hypothesis $$H_A:σ^2<1$$ at the $α=5\%$ significance level (you may also choose $α=1\%$ since it is not given by the exercise). The statistical test is $$χ^2=\frac{(n-1)S^2}{σ_0^2}\sim χ^2_{n-1}$$ where $σ_0=1.0.$ You will reject the null hupothesis if $$χ^2<χ^2_{n-1, \,α}$$