Question

You randomly select and measure the contents of 10 bottles of cough syrup. The results​ (in fluid​ ounces) are shown to the right.

4.213 4.293 4.252 4.242 4.183 4.293 4.265 4.243 4.226 4.235

Assume the sample is taken from a normally distributed population. Construct 95​%

confidence intervals for​ (a) the population variance

σ^2

and​ (b) the population standard deviation

σ.

Interpret the results.

Answer 1

	Values ( X )	Σ ( Xi - X̅ )2
	4.213	0.001
	4.293	0.0024
	4.252	0.0001
	4.242	0
	4.183	0.0038
	4.293	0.0024
	4.265	0.0004
	4.243	0
	4.226	0.0003
	4.235	0.0001
Total	42.445	0.0105

Mean X̅ = Σ Xi / n
X̅ = 42.445 / 10 = 4.2445

Sample Standard deviation SX = √ ( (Xi - X̅ )2 / n - 1 )
SX = √ ( 0.0105 / 10 -1 ) = 0.0342

χ2 (0.05/2, 10 - 1 ) = 19.0228
χ2 (1 - 0.05/2, 10 - 1) ) = 2.7004
Lower Limit = (( 10-1 ) 0.00117 / χ2 (0.05/2) ) = 0.0006
Upper Limit = (( 10-1 ) 0.00117 / χ2 (0.05/2) ) = 0.0039
95% Confidence interval is ( 0.0006 , 0.0039 )
( 0.0006 < σ2 < 0.0039 )
( 0.0235 < σ < 0.0624 )

We are 95% confidence that the true population variance lies within the interval  ( 0.0006 ,  0.0039 ).

We are 95% confidence that the true population standard deviation lies within the interval   ( 0.0235 , 0.0624 ).