Question

A textile fiber 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 Ho :u= 12 against H1 :u < 12, using a random sample of 4 specimens. Round your answers to 4 decimal places. (a) What is the type I error probability if the critical region is defined as ñ < 11.5 kilograms? The type I error probability is (b) Find ß for the case where the true mean elongation is 11.26 kilograms. B= (c) Find ß for the case where the true mean is 11.5 kilograms. B=

Answer 1

(a)

The z-score for $\bar{x}=11.5$ and $\mu=12$ is

$z=\frac{11.5-12}{0.5/\sqrt{4}}=-2$

The type I error is

$\alpha=P(z<-2)=0.0228$

(b)

Here we have o = 0.5, n =4 Hypotheses are: H: U H : Level of significance: 12 < 12 a =0.05 Since test is left tailed so critical values of z for which we will reject the null hypothesis is 2=-1.645. The critical value of sample mean for which we will reject the null hypothesis is: Xc = 2 +u =(-1.645*0.5/sqrt(4))+12 =11.589 The 2-score for: c = 11.589 and u = 11.26 Xc -11 = (11.589-11.26)/(0.5/sqrt(4)) oln = 1.32 The tyep Il error is the probability of fail to reject the false null hypothesis. So type Il error will be B =P(Z >= 1.32) =1-P(Z <= 1.32) =0.0934 Power of the test is: power = 1-B =0.9066 Excel function for critical value: Excel function used for beta: ENORMSINV(0.05) =1-NORMSDIST(1.32)

(c)

Level of significance: a =0.05 Since test is left tailed so critical values of z for which we will reject the null hypothesis is 2=-1.645. The critical value of sample mean for which we will reject the null hypothesis is: 0 Xc = 2 + 1 =(-1.645*0.5/sqrt(4))+12 n =11.589 The z-score for: Tc = 11.589 and = 11.5 z=*c = (11.589-11.5)/(0.5/sqrt(4)) olyn = 0.36 The tyep Il error is the probability of fail to reject the false null hypothesis. So type Il error will be B =P(Z >=0.36) =1-P( 2 <= 0.36) =0.3594 Power of the test is: power = 1-B =0.6406 Excel function for critical value: Excel function used for beta: ENORMSINV(0.05) =1-NORMSDIST(0.36)