## Related questions with answers

A rivet is to be inserted into a hole. A random sample of n = 15 parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measurements is s = 0.008 millimeters. Construct a 99% lower confidence bound for σ².

Solution

VerifiedIf $s^2$ is the sample variance from a random sample of $n$ observations from a normal distribution with unknown variance $\sigma^2$, then a $\textbf{$100(1 - \alpha)\%$ lower confidence bounds on $\sigma^2$}$ is

$\begin{align} \frac{(n-1)s^2}{\chi^2_{\alpha,n-1}}\leq\sigma^2 . \end{align}$

where $\chi^2_{\alpha,n-1}$ is the lower $100\alpha$ percentage points of the chi-square distribution with $n - 1$ degrees of freedom. We know that confidence level is $99\%$, degrees of freedom is $n-1= 15-1=14$ and sample standard deviation is $s=0.008$, than

$\begin{align*} \alpha=1-0.99=0.01\quad\Rightarrow\quad \chi^2_{0.01,14}= 29.14. \end{align*}$

Now from Equation (1) we get

$\frac{14\times0.008^2}{29.14}\leq\sigma^2.$

Therefore a lower confidence bound for $\sigma^2$

$\boxed{0.0000307\leq\sigma^2.}$

## Create a free account to view solutions

## Create a free account to view solutions

## Recommended textbook solutions

#### Probability and Statistics for Engineers and Scientists

9th Edition•ISBN: 9780321629111 (4 more)Keying E. Ye, Raymond H. Myers, Ronald E. Walpole, Sharon L. Myers#### Applied Statistics and Probability for Engineers

5th Edition•ISBN: 9780470053041Douglas C. Montgomery, George C. Runger#### Probability and Statistics for Engineers and Scientists

4th Edition•ISBN: 9781111827045 (2 more)Anthony J. Hayter#### Applied Statistics and Probability for Engineers

6th Edition•ISBN: 9781118539712 (1 more)Douglas C. Montgomery, George C. Runger## More related questions

1/4

1/7