Question

9. The lifetime of a certain brand of lightbulbs are assumed to be normally distributed. A researcher took a random sample of these lightbulbs and observed the following failure times: 17 | 17 | 24 24 | 25 34 40 Let xay be the critical value of a t random variable with v degrees of freedom. The following table lists values of x2., for specific combinations of a and v: v=6 v=7) a= 0.975 1.237 1.690 a = 0.95 1.635 1.2.167 a = 0.05 12.592 14.067 a= 0.025 14.449 16.013 Construct the 95% confidence interval for the variance of the light bulb lifetime.

Answer 1

A random sample size n = 7 is given.

17 , 17 , 24 , 24 , 25 , 34 , 40

n = 7

So, v = n - 1 = 6 (v denotes degrees of freedom)

We need to find sample variance s² of given sample.Using calculator

sample variance s² = 71.8095

Our aim is to construct 95% confidence interval.

$\therefore$ c = 0.95

$\therefore$$\alpha$ = 1- c = 1- 0.95 = 0.05

$\therefore$  $\alpha$/2 = 0.05 $\slash$ 2 = 0.025 and 1- $\alpha$ /2 = 0.975

$url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=VNlyKsXW6W7Iwynz5d_PIg--~C$_{$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=Xmon4xXLzqgT.OrOQi7.Jg--~C$ /2, v} = $url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=VNlyKsXW6W7Iwynz5d_PIg--~C$ _0.025,6 = 14.449

$url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=VNlyKsXW6W7Iwynz5d_PIg--~C$_{1-$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=Xmon4xXLzqgT.OrOQi7.Jg--~C$ /2,v} = $url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=VNlyKsXW6W7Iwynz5d_PIg--~C$ _0.975,6 = 1.237

The confidence interval for $\sigma$ ² is,

(n - 1)s² / $\chi$ ²_{$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=Xmon4xXLzqgT.OrOQi7.Jg--~C$/2} < $\sigma$ ² < (n - 1)s² / $\chi$ ²_{1 - $url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1573436422&ymreqid=50870fb5-0ab3-84d7-1ca9-bb00d101e500&sig=Xmon4xXLzqgT.OrOQi7.Jg--~C$ /2}

( 6 * 71.8095) / 14.449 < $\sigma$ ² < ( 6 * 71.8095) / 1.237

29.8191  < $\sigma$ ² < 348.3080

Required interval is (29.8191 , 348.3080 )