4. A bearing ball manufacturer is under the assumption that they are producing b

Question

4. A bearing ball manufacturer is under the assumption that they are producing bearing balls of diameter 60 um. Of course the actual diameters are normally distributed with mean 60 and variance of 16. Recently there are more than usual number of boxes of the bearing returned to the factory as the diameters exceeded certain threshold level. we suspect that our machines may be out of calibration and the mean of diameters may have gradually exceeded 60um. a) We take a sample of n 100 bearings to run a test of hypothesis. Determine the 'test statistic' for this hypothesis testing, and the critical values related to the significance level 0.05. b) Suppose the sample mean is 60.5 pm. What is the conclusion of the test? 5. Back to question 4; this time we don't trust our knowledge of the variance, and we calculate and use the sample variance instead. We take a smaller sample, n 3 81 and calculate S 3.8. Repeat parts a and b of question 1, and after doing part b mention the type of error we committed. (this time also the sample average is 60.5.)

Explanation / Answer

Part a)

Here the population standard is known so this is the case of z-test for single mean.

Here the claim says that it is one-tailed test (right tailed)

The level of significance is 5%. From normal table we get the critical value as 1.645

Part b)

Null and Alternative Hypothesis:

H0: µ 60

H1: µ > 60

Level of significance (a) = .05

Test Statistics:

z = ( x bar - µ ) / ( s / sqrt (n))

= (60.5 -60)/(4 / sqrt(100))                [ Variance = 16, so s = sqrt (16) = 4 ]

= 1.25

Decision: The Null hypothesis can be rejected if the calculated test statistics is greater or equal to the critical value of 1.645.

Here we find the value of the test statistics 1.25 is less than the critical value of 1.645; we fail to reject the null hypothesis.

Conclusion:

At 5% level of significance there is not sufficient evidence to conclude that the mean diameter may have gradually exceeded.

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