Find the 95% confidence interval for the standard deviation of the lengths of pipes if a sample of 11 pipes has a standard deviation of 10.6 inches.
A) $54.9

Question

Find the 95% confidence interval for the standard deviation of the lengths of pipes if a sample of 11 pipes has a standard deviation of 10.6 inches. 
A) $54.9 < \sigma < 346.0$
B) $7.6 < \sigma < 9.6$
C) $7.4 < \sigma < 18.6$
D) $101.4 < \sigma < 123.4$

Quizplus · Accepted Answer

To find the 95% confidence interval for the standard deviation, we use the chi-square distribution because the standard deviation is a measure of spread and not central tendency. 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 level of confidence. The degrees of freedom here are 11 - 1 = 10. The chi-square values for 95% confidence and 10 degrees of freedom are approximately 3.94 (lower) and 20.48 (upper). The confidence interval for the population standard deviation ($\sigma$) is calculated using the formula: $$s\sqrt{\frac{n-1}{\chi^2_{\alpha/2}}}<\sigma<s\sqrt{\frac{n-1}{\chi^2_{1-\alpha/2}}}$$ Substituting the given values and the chi-square values, we get the interval for $\sigma$ which matches the range in option C, indicating it is the correct choice.

Find the 95% Confidence Interval for the Standard Deviation of the Lengths