Question

In: Statistics and Probability

A factory supplies parts for motorcycles. It produces a cylindrical engine part that is supposed to...

A factory supplies parts for motorcycles. It produces a cylindrical engine part that is supposed to have a diameter of 6 centimetres. The factory’s quality control manager suspects that the machine producing the engine part is not working properly and that the mean diameter being produced is less than the correct size. To test if this belief is correct, a random sample of 40 parts is tested and measured. The sample mean diameter was found to be 5.9 cm. Assume that diameter size of the engine part is normally distributed with a population standard deviation of 0.26 cm.

  1. You were recently hired as a statistician working for the factory. Assist the quality control manager in performing a hypothesis test at a 3% level of significance to verify his belief. Display the six steps process (involving drawing the rejection region/s and determining the critical value/s for the decision rule) in performing the test.

  1. Calculate the p-value of the test above. Display working. State the decision rule should you want to use the p-value method in doing hypothesis testing.  
  2. Now that you have performed the test in part a), would you suggest to the manager

to replace the machine? Why?   

  1. Identify which one of these two types of error (Type I or Type II) you could make with

the conclusion you made in part a). Briefly explain your selection.

Solutions

Expert Solution

a)Ho :   µ =   6                  
Ha :   µ <   6       (Left tail test)          
                          
Level of Significance ,    α =    0.03                  
population std dev ,    σ =    0.2600                  
Sample Size ,   n =    40                  
Sample Mean,    x̅ =   5.9000                  
                          
'   '   '             

critical z value, z* =       -1.8808   [Excel formula =NORMSINV(α/no. of tails) ]      

rejecction region :

|test stat| > |1.8808|      
                          
Standard Error , SE = σ/√n =   0.2600   / √    40   =   0.0411      
Z-test statistic= (x̅ - µ )/SE = (   5.900   -   6   ) /    0.0411   =   -2.43
                          

                             
Decision:   |test stat| > |critical value | ,  Reject null hypothesis

.............

b)

p-Value   =   0.0075   [ Excel formula =NORMSDIST(z) ]
Decision:   p-value<α, Reject null hypothesis       

..........

c)

the machine producing the engine part is not working properly and that the mean diameter being produced is less than the correct size

.............

d)

type I errro is there

....................

THANKS

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