A machine that grinds valves is set to produce valves whose lengths have mean 10

Question

A machine that grinds valves is set to produce valves whose lengths have mean 100 mm and standard deviation 0.1 mm. The machine is moved to a new location. It is thought that the move may have upset the calibration for the mean length, but that it is unlikely to have changed the standard deviation. Let µ represent the mean length of valves produced after the move. To test the calibration, a sample of 100 valves will be ground, their lengths will be measured, and a test will be made of the hypotheses H0: µ=100 versus H1: µ100. Find the rejection region if the test is made at the 5% level? If the sample mean length is 99.97 mm, will H0 be rejected at the 5% level?

Explanation / Answer

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   100  
Ha:    u   =/   100  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical z, as alpha =    0.05   ,      
alpha/2 =    0.025          
zcrit =    +/-   1.959963985      

Hence, reject Ho when z < -1.96 or z > 1.96. [ANSWER, REJECTION REGION]

********************************************
              
Getting the test statistic, as              
              
X = sample mean =    99.97          
uo = hypothesized mean =    100          
n = sample size =    100          
s = standard deviation =    0.1          
              
Thus, z = (X - uo) * sqrt(n) / s =    -3          
              
Also, the p value is              
              
p =    0.002699796          
              
As |z| > 1.96, and P < 0.05, we   REJECT THE NULL HYPOTHESIS.          

Hence,there is significant evidence that the population mean is not 100 mm at 0.05 level. [CONCLUSION]

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