A parachute company sells safety devices that are designed to fire, releasing the reserve parachute, at a mean height of 304.8 m with a standard deviation of 11.2 m. The company recently updated the devices, which, the lead engineer suspects, has caused the standard deviation, $\sigma$, to decrease. He measures the heights at which the devices fire for a random sample of 10 of the updated devices. The heights at which the devices fire have a sample standard deviation of 7.6 m. Assume the heights at which the updated devices fire follow a normal distribution.
Can the engineer conclude that the population standard deviation, $\sigma$, is less than 11.2 m? To answer, complete the parts below to perform a hypothesis test. Use the 0.10 level of significance.
(a) State the null hypothesis $H_0$ and the alternative hypothesis $H_1$ that you would use for the test.
(b) Perform a chi-square test and find the $p$-value.
Here is some information to help you with your chi-square test.
$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$
The value of the test statistic is given by $\chi^2 = \frac{(n-1)s^2}{\sigma^2}$.
The $p$-value is the area under the curve to the left of the value of the test statistic.
(c) Based on your answer to part (b), choose what can be concluded, at the 0.10 level of significance, about the population standard deviation in the heights at which the updated devices fire.
