# Statistics Questions

Subject Mathematics General Statistics

## Question

1. In its current configuration, an assembly line produces an average of 800 personal computers per day with standard deviation 25 computers. A newly hired production engineer believes that by switching from a "push" assembly line approach to a "pull" (JIT) approach, the process can be speeded up with no compromise in quality of assembly. To test this belief, an experimental assembly line is set up for n days using the pull approach, and we observe the number of computers produced each day. Assume that the standard deviation for the experimental line is the same as for the existing line, and that the number of computers produced per day is approximately normally distributed. The question under study is whether the results from the experiment provide strong evidence that the new approach increases daily output.

(a) State the appropriate null and alternative hypotheses.
(b) What are the implications, to the firm, of making a Type I error in this hypothesis testing situation? Don’t just repeat the definition of a Type I error. Tell me a (short) story!
(c) Repeat Part (b) for a Type II error.
(d) Suppose that the Type I error probability is specified as 0.05, and that the sample size is n = 50. State the decision rule for testing the hypothesis that you specified in Part (a) (i) in terms of the z-statistic; (ii) in terms of the sample mean ; and (iii) in terms of the p-value.
(e) The test is conducted (using n = 50) and the result is 804 units. Report the test conclusion. Show the numbers/logic used to reach this conclusion and interpret it in the context of the problem. Should the company switch to the ‘pull’ system?

2. Refer to the scenario of Problem 3, and suppose that we’re back at the test design stage so the appropriate sample size has not yet been determined and consequently no data has been obtained. Continue to assume a known standard deviation of 25 and use level of significance 0.05.

(a) An increase of 10 units per day in the line’s average production rate is considered to be very important. What sample size should be used if the test is to have an 80% chance of detecting such an increase? A 90% chance? a 95% chance?
(b) Suppose that we decide to base the test on a sample of size n = 50. Calculate the power of the test at µ = 800.01 and for µ-values ranging from 802.5 to 825 in increments of 2.5. Report your results in a table. Then create a pretty power function plot (power in the vertical axis vs. µ in the horizontal axis).

## Solution Preview

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1.

(a)
H0: mu = 800
H1: mu > 800

(b)
It takes a risk of switching to a pull approach which is actually not as productive as the current method.

(c)
It takes a risk of continuing to use the current method while the JIT approach actually performs better.

(d)
n=50
(i) Reject if z = n1/2 ( x ̅ – mu )/σ > 1.645
(ii) Reject if x ̅ >1.645 σ/sqrt(n) + mu
(iii) Reject if p-value is less than 0.05...

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