1. The load life X was measured (in millions of revolutions) for 80 ceramic ball
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Question
1. The load life X was measured (in millions of revolutions) for 80 ceramic ball bearings subjected to a 6.45kN load. A summary of the data from JMP is on the next page a) The load life needs to average more than 150 for bearings of this type. Is there sufficient evidence to justify this claim? Explain fully, showing all parts of a hypothesis test, with 0.01 significance level b) What do the normal diagnostic plot (Q-Q plot) and Shapiro-Wilk's test indicate about distribution of X, in terms of shape, symmetry, etc.? Is a normal distribution plausible? c) Use the delta method to approximate SE(X1/) Summary Statistics Mean Std Dev Goodness-of-Fit Test Shapiro-Wilk W Test 175.6242 86.289831 80 7445.9349 1.0461532 1.0105432 49.133224 62.52666 450.14769 158.04947 P-Value 0.0001 0.921522 Variance Skewness Kurtosis CV Minimum Maximum Median Normal Diagnostic Plot 500 450 400 350 300 250 200 150 100 50 0015 0.05 0.09 0.16 0.3 0.5 0.7 0.84 0.91 0.95Explanation / Answer
1) as n = 80 , hence we will conduct a z test
so mean = 175.62 and SD = 86.28 , n = 80
so we calculate the z stat as
(X-mean)/(sd/sqrt(n)) = (150-175.62)/(86.28/sqrt(80)) = -2.655
please keep the z tables ready to check the probability as
P ( Z<2.655 )=1P ( Z<2.655 )=10.9961=0.0039
as the p value is less than 0.01
hence we can conclude that there is enough evidence to claim that the mean is greater than 150
b)
as the dat apoints in the qqplot follows quite close to the tangent line , we can conclude that the data distribution is quite close to normal distribution
same is evident from the p value of the shapiro wilk test. as the p value is 0.0001 , which is less than 0.01 , hence we can conclude that the normality check suggests that data comes from a normal distribution
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