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9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 297

Using the Computer
Many statistics software packages will calculate sample sizes and type II error probabilities. To
illustrate, here are some computations from Minitab for the propellant burning rate problem.

Power and Sample Size
1-Sample Z Test
Testing mean null (versus not null)
Calculating power for mean null + difference
Alpha 0.05 Sigma 2
Sample Target Actual
Difference Size Power Power
1 43 0.9000 0.9064
Power and Sample Size
1-Sample Z Test
Testing mean null (versus not null)
Calculating power for mean null difference
Alpha 0.05 Sigma 2
Sample Target Actual
Difference Size Power Power
1 28 0.7500 0.7536

Power and Sample Size
1-Sample Z Test
Testing mean null (versus not null)
Calculating power for mean null difference
Alpha 0.05 Sigma 2
Sample
Difference Size Power
1 25 0.7054

In the first part of the boxed display, we asked Minitab to work Example 9-3, that is, to find
the sample size n that would allow detection of a difference from 50 of 1 centimeter per
0
second with power of 0.9 and 0.05. The answer, n 43, agrees closely with the calcu-
lated value from Equation 9-19 in Example 9-3, which was n 42. The difference is due to
Minitab using a value of z that has more than two decimal places. The second part of the com-

puter output relaxes the power requirement to 0.75. Note that the effect is to reduce the
required sample size to n 28. The third part of the output is the solution to Example 9-4,
where we wish to determine the type II error probability of ( ) or the power 1
for the
sample size n 25. Note that Minitab computes the power to be 0.7054, which agrees closely
with the answer obtained from the O.C. curve in Example 9-4. Generally, however, the com-
puter calculations will be more accurate than visually reading values from an O.C. curve.

9-2.5 Large-Sample Test

We have developed the test procedure for the null hypothesis H : assuming that the pop-
0
0
2
ulation is normally distributed and that is known. In many if not most practical situations 2