Question

. Two machines are used to fill plastic bottles with dish-washing detergent. The standard deviations of fill volume are known to be σ1 = 0.10 and σ2 = 0.15 fluid ounces for the two machines, respectively. Two random samples of n1 = 12 bottles from machine 1 and n2 = 10 bottles from machine 2 are selected, and the sample mean fill volumes are ¯ x1 = 30.61 and ¯ x2 = 30.34 fluid ounces. Assume normality and independence.

(a) Is there evidence at the 10% level to claim that the two machines fill the bottles to different volumes on average?

(b) Construct a 90% two-sided CI on the mean difference in fill volume.

Answer 1

To Test :-
H0 :- µ1 = µ2
H1 :- µ1 ≠ µ2

Test Statistic :-

Z = (X1 - X2)/V ( nl) + (o/n2)


Z = (30.61 – 30.34)/ (0.12/12) + (0.152/10)

Z = 4.8624 ≈ 4.86

Test Criteria :-
Reject null hypothesis if | Z | > Z( α/2)
Critical value Z(α/2) = Z(0.1 /2) = 1.645 ( From standard normal table )
| Z | > Z(α/2) = 4.8624 > 1.645
Result :- Reject Null Hypothesis

Part b)
Confidence interval :-

(#1 – X2) – Za/2V (07/n1)+(cz/n2)

Z(α/2) = Z (0.1 /2) = 1.645

(30.61 – 30.34) 20.1/2 (0.12/12) + (0.152/10)

Lower Limit =
(30.61 – 30.34) – 20.1/2 (0.12/12) + (0.152/10)

Lower Limit = 0.1787
Upper Limit =
(30.61 – 30.34) + Z01/2V (0.12/12) + (0.152/10)

Upper Limit = 0.3613
90% Confidence interval is ( 0.1787 , 0.3613 )