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Let $Y1,Y2,...,Y11$ be iid $N(μ,σ^2)$ so that n= 11. Find $P(0.325<S^2/σ^2<2.048)$

My approach was: $P(S^2/σ^2<2.048) - P(S^2/σ^2<0.325)$

The degrees of freedom would be 10. This is the table I'm attempting to use.

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However, I'm not entirely sure where to go from here. I did use R software to get the solution of 0.004040321 - but I don't know if this is correct.

Jones
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1 Answers1

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You have to convert your random variable to a $\chi^2_{n-1}$ random variable. Recall $$\dfrac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$$ so you have to write $$\mathbb{P}\left(\dfrac{S^2}{\sigma^2} < 2.048 \right) = \mathbb{P}\left(\dfrac{(11-1)S^2}{\sigma^2} < 2.048 \cdot (11-1)\right) = \mathbb{P}\left(\chi^2_{10} < 20.48\right)$$ and similarly, $$\mathbb{P}\left(\dfrac{S^2}{\sigma^2} < 0.325 \right) = \mathbb{P}(\chi^2_{10} < 3.25)\text{.}$$ Can you finish this up?

Clarinetist
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  • Sorry to bother you, but I would like to respectfully remind you to award the full bounty to the answer https://math.stackexchange.com/a/4145484/64809. I have checked and accepted that answer. – Hans May 26 '21 at 17:28
  • @Hans No problem, thank you for letting me know and I've awarded the bounty. – Clarinetist May 26 '21 at 17:51