Answer to Question #235511 in Statistics and Probability for blossomqt

Question

  The percentage of titanium in an alloy used in aerospace castings is measured in 51 randomly selected parts. The sample standard deviation is s = 0.37. Construct a 95% two-sided confidence interval for the standard deviation

Answer 1

QUESTION

 The percentage of titanium in an alloy used in aerospace castings is measured in 51 randomly selected parts. The sample standard deviation is s = 0.37. Construct a 95% two-sided confidence interval for the standard deviation.

SOLUTION

The confidence interval for population standard deviation is given by:

$\sqrt[s]{\frac{n-1}{X{2\atop \alpha/2,n-1}}} \lt \sigma \lt \sqrt[s]{\frac{n-1}{X{2\atop \alpha/2,n-1}}}$

having been given the:

Significance level : $\alpha = 1-0.95 = 0.05$

Sample size : $n = 51$

Sample standard deviation : $s = 0.37$

Then we use the chi-square distribution table, we get:

$X{2\atop 1-\alpha/2,n-1} = X{2\atop 0.975,50} = 32.36$

$X{2\atop \alpha/2,n-1} = X{2\atop 0.025,50} = 71.42$

Confidence interval for population standard deviation will be:

$\sqrt[0.37]{\frac{50}{32.36}} \lt \sigma \lt \sqrt[0.37]{\frac{50}{71.42}}$

$0.45992018426 \lt \sigma \lt 0.30958278534$

ANSWER: $0.45992018426 \lt \sigma \lt 0.30958278534$