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: $0.45992018426 \lt \sigma \lt 0.30958278534$