A textile fiver manufacturer is investigating a new drapery yarn, which the comp

Question

A textile fiver manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. The company wishes to test the hypothesis H0: mu=12 against H1: mu less than 12 using a random sample of n specimens. Find the boundary of the critical region if the type I error probability is given below. Let x bar c represent the critical value for the mean.

a.) alpha = 0.03 and n=4; x bar c = ?

b.) alpha =0.05 and n=4; x bar c =?

c.) alpha=0.03 and n=16; x bar c=?

d.) alpha=0.05 and n=16; x bar c=?

Explanation / Answer

Solution:

a) t(0.03, 3df) = 2.950
b) t(0.05, 3df) = 2.353
c) t(0.03, 15df) = 2.034
d) t(0.05, 15df) = 1.753

The t-statistic is:
(xbar-12)/(sd/sqrt(n))
reject H0 when t < critical value or xbar < 12 - tcrit(sd/sqrt(n))

a) (x-12)/(0.5/sqrt(4)) = 2.950   
=> x = (2.950*0.25) +12 = 12.7375
b) (x-12)/(0.5/sqrt(4)) = 2.353
=> x = (2.353*0.25) +12 = 12.58825
c) (x-12)/(0.5/sqrt(4)) = 2.034   
=> x = (2.034*0.25) +12 = 12.5085
d) (x-12)/(0.5/sqrt(4)) = 1.753
=> x = (1.753 *0.25) +12 = 12.43825

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