To find the standard deviation of the diameter of wooden dowels, the manufacturer measures 19 randomly selected dowels and finds the standard deviation of the sample to be $s = 0.16$. Find the $95 \%$ confidence interval for the population standard deviation $\sigma$.
A) $0.12

Question

To find the standard deviation of the diameter of wooden dowels, the manufacturer measures 19 randomly selected dowels and finds the standard deviation of the sample to be $s = 0.16$. Find the $95 \%$ confidence interval for the population standard deviation $\sigma$.
A) $0.12 < \sigma < 0.24$
B) $0.15 < \sigma < 0.21$
C) $0.11 < \sigma < 0.25$
D) $0.13 < \sigma < 0.22$

Quizplus · Accepted Answer

The formula for the confidence interval for the population standard deviation is:$$\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}} < \sigma^2 < \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}$$where $s$ is the sample standard deviation, $n$ is the sample size, and $\chi^2_{\alpha/2, n-1}$ and $\chi^2_{1-\alpha/2, n-1}$ are the chi-square critical values with $n-1$ degrees of freedom and significance levels $\alpha/2$ and $1-\alpha/2$, respectively.For a 95% confidence level, $\alpha = 0.05$, so $\alpha/2 = 0.025$ and $1-\alpha/2 = 0.975$. Using a chi-square table, we find that $\chi^2_{0.025, 18} = 10.83$ and $\chi^2_{0.975, 18} = 30.14$.Plugging in the values, we get:$$\frac{(19-1)0.16^2}{30.14} < \sigma^2 < \frac{(19-1)0.16^2}{10.83}$$$$\Rightarrow 0.12 < \sigma < 0.24$$Therefore, the correct answer is A.

To Find the Standard Deviation of the Diameter of Wooden s=0.16s = 0.16s=0.16

To Find the Standard Deviation of the Diameter of Wooden $s = 0.16$