The engine camshaft of a Toyota Camry is a vital part and it must have the appro

Question

The engine camshaft of a Toyota Camry is a vital part and it must have the appropriate hardness to wear properly as it has repeated contact with the engine lifters. To harden the camshaft, Toyota used an existing heat treatment of the shafts and they look a random sample of size 28 and found that the camshaft hardness depth mean and standard deviations to be X1 = 4.3mm and S1 = 0.47 mm respectively. It is known the heat treatment process is normally distributed. Toyota developed a new heating process to reduce the variations in the camshaft hardness depth and used it. Toyota took a random sample of 30 camshafts after heating them using the new heating process and found that the camshaft hardness depth mean and standard deviations to be X2 = 4.5mm and S2 = 0.297489 mm respectively. Does the data support the objective of Toyota that the population variance of all camshafts has been reduced by using the new heating process? Perform a Hypothesis test: list all the required 5 steps in order to answer the question. DO NOT use the P-VALUE or the Confidence interval as a rejection rule here.

Explanation / Answer

Step 1: State the Null Hypothesis and the Alternative Hypothesis.
H0 : 12 = 22   ag.   H1 : 12 > 22,
where 12 , 22 are the population variances of all camshafts before and after the new heating process respectively.

Step 2 : Set the level of significance : = 0.05

Step 3 : Calculate the test statistic.

Fcalculated = s12 /s22 , where s12 ,s22 are the sample variances.

               = 2.4960535

Step 4: Construct Acceptance / Rejection regions.

Fcritical = F,N11,N21 , where F,N11,N21 is the upper 100% point of F-distribution with N1-1 and N2-1 degrees of freedom
= 1.875131

Critical region : Fcalculated > Fcritical .

Step 5 : Conclusion about H0.

Since Fcalculated > Fcritical , we reject H0 at 5% level of significance and conclude that the population variances of all camshafts has been reduced after the new heating process.

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