You are testing a new package handling system. The Historical average is 35 minutes with a population standard deviation of 8 minutes. A test of the new process on 10 random runs has a mean of 33 minutes. Let’s say 33 was in fact the true mean of the alternative hypothesis – in other words an infinite number of sample means at the new process would have resulted in a mean of 33? What is the beta error and power of this experiment? (note: this one is difficult. Draw a picture of the alternative hypothesis)
A) 
Beta 50%, Power 50% 

B)Beta 20%, Power 80% 

C)Beta 80%, Power 20% 

D)Beta 40%, Power 40% 
Same Problem: It will likely cost the company $20,000 to change the process. Management likes the opportunity at the alternative hypothesis (the 33 vs. 35 minutes) but does not like your conclusion. What might be your next step?
A)Approximately two minutes away from the critical value at an alpha level of 5% and there is not enough evidence to reject it at 33 minutes. Do not reject the null hypothesis. Best to resign my position if management decides to proceed with the change. 

B)Change the test. Increase n from 10 to 15 and rerun it. 

C)The increase in the Power level will come from reducing the Alpha error. Change the confidence level from 95% to 70%. (note: if you picked this one, please see the second part of answer a). 

D)Rerun the test but question not just the sample size but the value of the population standard deviation. How is it measured? Is it all product lines? Is it across all environmental conditions such as humidity and temperature? Is it across different operators? Is it in control? What can be done to reduce it? 
i) Here we're testing:
, thus at a significance level of 0.05, we reject the null hypothesis if the Zscore of the obtained mean is less than 1.645. Here the sample mean follows a normal distribution with mean 35 and standard deviation. Thus the critical region is mean less than 30.839.
Now under the alternate hypothesis sample mean follows a normal distribution with mean 35 and standard deviation. Hence the power of the test is given by:
. Therefore power is 20% while Beta=80%
ii) B)Change the test. Increase n from 10 to 15 and rerun it. This is because increasing the sample size will make our test result better.
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