A random sample of 15 crates have a mean weight of $165.2$ pounds and a standard deviation of $13.5$ pounds. Construct a $95 \%$ confidence interval for the population standard deviation $\sigma$. Assume the population is normally distributed, and round to the nearest hundredth when necessary.
A) $( 9.88,21.29 )$
B) $( 10.38,19.71 )$
C) $( 97.69,453.3 )$
D) $( 2.69,5.79 )$

Question

Quizplus · Accepted Answer

To construct a 95% confidence interval for the population standard deviation, we use the chi-square distribution because the sample size is small and the population is assumed to be normally distributed. The formula involves the sample standard deviation $s$, the sample size $n$, and the chi-square values for the degrees of freedom $n-1$ at the desired confidence level. The confidence interval is given by: $\sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}}$ to $\sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}}$, where $\chi^2_{\alpha/2}$ and $\chi^2_{1-\alpha/2}$ are the chi-square values for $\alpha/2$ and $1-\alpha/2$, respectively, with $n-1$ degrees of freedom. Plugging in the given values and calculating, option A provides the correct interval for the population standard deviation.

A Random Sample of 15 Crates Have a Mean Weight 165.2165.2165.2

A Random Sample of 15 Crates Have a Mean Weight $165.2$