Question

certain automatic machine center set to produce pin parts that will have a mean diameter of 40 mm with a variance of 0.0001 mm² to be used in a certain engineering application. A sample of five parts were measured and found for this sample the mean diameter is 4.0001 cm and the variance is 0.00090601 mm². Test the appropriate hypothesis that the machine is in proper working order using a risk of wrong judgement of

1. 10%   
2. 1%

Answer 1

We are to test if the machine is in proper working condition. Thus we are to test that the difference between the pre set mean population diameter and that obtained from the observed population are equal or not. Let the mean population diameter obtained from the observed population be . Thus the null and alternative hypotheses are given by

Here we perform 1 sample z test . We summarize the given data as

Sample size	n=5
Sample mean	=40.001
Population SD

The test statistic is given by

The test statistic under the null hypothesis follows N(0,1). The critical valuesas obtained from the Biometrika table is given by

As the observed value is less than both the critical values, we fail to reject the null hypothesis at 1% and 10% level of significance and thus we can conclude that the process mean is in control.

Again we are to test whether the pre set process variability and that observed differs or not. Let the population process variability that is observed be . Thus the null and alternative hypotheses are given by

Here we perform 1 sample z test . We summarize the given data as

Sample size	n=5
Population mean	=40
Sample variance

The test statistic is given by

The test statistic under the null hypothesis follows chi square distribution with df n-1=4.

The critical values as obtained from the Biometrika table are given by

and

It is evident that observed value is greater than the critical values,the null hypothesis is rejected and hence we can conclude that the process variability is not in control at both 1% and 10% level of significance.

CONCLUSION:

The process central tendency is in control but not the process variability and thus the process is out of control.

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