Question

The diameters​ (in inches) of

1717

randomly selected bolts produced by a machine are listed. Use a

9595​%

level of confidence to construct a confidence interval for ​(a) the population variance

sigma squaredσ2

and ​(b) the population standard deviation

sigmaσ.

Interpret the results.

4.4684.468	4.4364.436	4.0234.023	4.3134.313	4.0064.006	3.7783.778
3.8283.828	3.7473.747	4.2374.237	3.9393.939	4.1274.127	4.5734.573
3.9743.974	3.7573.757	3.8733.873	3.8143.814	4.4294.429

​(a) The confidence interval for the population variance is

Answer 1

	Values ( X )	Σ ( Xi- X̅ )2
	4.468	0.1523
	3.828	0.0624
	3.974	0.0108
	4.436	0.1283
	3.747	0.1094
	3.757	0.1029
	4.023	0.003
	4.237	0.0253
	3.873	0.0419
	4.313	0.0553
	3.939	0.0193
	3.814	0.0696
	4.0	0.0052
	4.127	0.0024
	4.429	0.1233
	3.778	0.0899
	4.573	0.2452
Total	69.322	1.2465

Mean X̅ = Σ Xi / n
X̅ = 69.322 / 17 = 4.0778
Sample Standard deviation S_X = √ ( (Xi - X̅ )² / n - 1 )
S_X = √ ( 1.2465 / 17 -1 ) = 0.2791

χ² (0.05/2) = 28.8454
χ² (1 - 0.05/2) ) = 6.9077
Lower Limit = (( 17-1 ) 0.0779 / χ² (0.05/2) ) = 0.0432
Upper Limit = (( 17-1 ) 0.0779 / χ² (0.05/2) ) = 0.1804
95% Confidence interval is ( 0.0432 , 0.1804 )

Part a)
( 0.0432 < σ² < 0.1804 )

We are 95% confident that the true population variance lies within the interval.

Part b)

( 0.2079 < σ < 0.4248 )

We are 95% confident that the true population standard deviation lies within the interval.