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.