A manufacturer claims that a new fiber optic cable has a mean breaking strength

Question

A manufacturer claims that a new fiber optic cable has a mean breaking strength = 15 kg with a standard deviation of 0.5 kg. To test the null hypothesis that = 15 kg against the alternative that < 15 kg, a random sample of lengths from 50 different rolls of cable will be tested. The test will be conducted with = 0.05.


a) Define Type I error and explain what it means in the context of drawing a conclusion from sample evidence and explain what are the consequences of reducing = 0.05 to = 0.01 for a test of hypothesis

Fiber Cable Strength

14.708
14.463
14.056
14.718
14.510
14.586
14.402
15.172
13.922
14.442
14.743
15.626
15.061
14.707
13.151
14.431
13.937
14.615
15.634
15.340
15.284
14.872
14.039
14.851
15.054
14.456
14.666
15.110
15.335
15.500
14.070
15.824
13.702
14.621
15.187
15.003
14.748
14.314
15.750
14.825
15.457
14.527
14.976
15.115
14.511
14.494
14.603
15.093
14.295
14.931

Explanation / Answer

Ans:

sample size=n=50

sample mean=14.75

population standard deviation=0.5

here,sample size,n>=30,we can use z statistic.

Test statistic:

z=(14.75-15)*sqrt(50)/0.5=-3.54

p-value=P(z<-3.54)=0.0002

As,p-value <0.05,we reject null hypothesis.

There is suffcient evidence to conclude that new fiber optic cable has a mean breaking strength is less than 15 kg.

Type I error:when we reject null hypothesis and it is actually true.

when we reject null hypothesis that mean breaking strength is equal to 15 kg and actually breaking strenght is equal to 15 kg,we will make type I error.

When alpha is reduced to 0.01,we will be less likely to rejct null hypothesis(as we reject H0,when p- value<alpha) i.e probability of accepting null hypothesis increases,probability of type I error reduces.

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