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A gun-like apparatus has recently been designed to replace needles in administering vaccines. The apparatus can be set to inject different amounts of the serum, but because of random fluctuations the actual amount injected is normally distributed with a mean equal to the setting and with an unknown variance $\sigma^2$. It has been decided that the apparatus would be too dangerous to use if $σ$ exceeds $0.10$. If a random sample of $50$ injections resulted in a sample standard deviation of $0.08$, should use of the new apparatus be discontinued?

Comment on the appropriate choice of a significance level for this problem, as well as the appropriate choice of the null hypothesis

I approached this question as follows:

null hypothsis $H_0: \sigma \le 0.10 $ and alternate hypothesis $H_1: \sigma > 0.10 $.

Then I used Chi square: $\chi_0^2 = \frac{(n-1)(s^2)}{\sigma^2} = \frac{(49)(.0064)}{.01} = 31.36$.

I calculated the probability that a Chi Square variable with $49$ degrees of freedom would have a $97.6513\%$ chance of being above $31.36$: $P\{\chi_{49}^2 > 31.36\} = .976513$.

What exactly does this mean? Does it mean that if $\sigma$ was to equal $.10$, there would be a $97.6513\%$ of finding a sample of 50 that would give you a sample standard deviation of $.08$? Also, I'm having doubts that I used the correct null hypothesis since this question seems too obvious as it stands.

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I think you meant to write the alternate hypothesis as $H_1$. Also, the unknown variance should be $\sigma^2$ rather than $\sigma$. – Sarastro Dec 3 '12 at 4:00
Also $10$ is presumably meant to be $0.10$? – joriki Dec 3 '12 at 4:14
up vote 2 down vote accepted

The alternative hypothesis is what requires a proof in such a situation, so you should use $H_0: \sigma \ge 0.10$ (not $10$) and $H_1: \sigma < 0.10$.

Your $\chi^2$ calculation seems to be correct. What you have shown is that if the true standard deviation were 0.1 (the minimum unsafe level), then observing a sample standard deviation of 0.08 or smaller has probability $1-0.976... \approx 0.0235$. That is, the observed sample standard deviation is not really consistent with the hypothesis that the true standard deviation is 0.1 or larger.

So you have evidence that the true standard deviation is < 0.1 and that the device is safe.

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Shouldn't the null hypothesis be $H_0: \sigma > 0.10$ and alternate be $H_1: \sigma \le 0.10$? I ask this because the device is still safe if it is equal to $0.10$ - it's only dangerous if it exceeds 10. – CodyBugstein Dec 3 '12 at 17:11
It won't ever make a difference in the decision. – Hans Engler Dec 3 '12 at 17:17

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