Applied Statistics and Probability for Engineers

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.

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

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 nthat would allow detection of a difference from  0 50 of 1 centimeter per

second with power of 0.9 and    0.05. The answer, n43, agrees closely with the calcu-

lated value from Equation 9-19 in Example 9-3, which was n42. The difference is due to

Minitab using a value of zthat 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 n28. 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 n25. 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 0 :   0 assuming that the pop-

ulation is normally distributed and that 2 is known. In many if not most practical situations 2

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