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Sample Size Formulas It is also possible to obtain formulas for calculating the sample sizes directly. Suppose that the null hypothesis H0: 1 , 2 0 is false and that the true difference in means is 1 2 . One may nd formulas for the sample size required to obtain a speci c value where 0 of the type II error probability for a given difference in means and level of signi cance .

For the two-sided alternative hypothesis with signi cance level , the sample size n1 n2 n required to detect a true difference in means of with power at least 1 is n 1z

+ F(Qi-1,

z 221

0 2 1n 2

Q i - 2 , Qi-3)

2 22

(10-5)

22 2

= Qi-1

For a one-sided alternative hypothesis with signi cance level , the sample size n1 n2 n required to detect a true difference in means of ( 0) with power at least 1 is n 1z 1 z 221

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2 1 02 2 22

= Qi-1

(10-6)

The derivation of Equations 10-5 and 10-6 closely follows the single-sample case in Section 9-2.3. For example, to obtain Equation 10-6, we rst write the expression for the -error for the two-sided alternate, which is z B n1 B n1

+ ( ( Q i - 4 + H(Qi-1, Qi-2,

0 2 2

0 2 2

where is the true difference in means of interest. Then by following a procedure similar to that used to obtain Equation 9-17, the expression for can be obtained for the case where n n1 n2. EXAMPLE 10-3 To illustrate the use of these sample size equations, consider the situation described in Example 10-1, and suppose that if the true difference in drying times is as much as 10 minutes, we want to detect this with probability at least 0.90. Under the null hypothesis, 0 0. We have a one-sided alternative hypothesis with 10, 0.05 (so z z0.05 1.645), and since the power is 0.9, 0.10 (so z z0.10 1.28). Therefore we may nd the required sample size from Equation 10-6 as follows: n 1z 1 z 221

Qi-3)

2 1 02 2 22

1.282 2 3 182 2 110 02 2

((Qi-4

182 2 4

10-2.3

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Engineers and scientists are often interested in comparing two different conditions to determine whether either condition produces a signi cant effect on the response that is observed These conditions are sometimes called treatments Example 10-1 illustrates such a situation; the two different treatments are the two paint formulations, and the response is the drying time The purpose of the study is to determine whether the new formulation results in a signi cant effect reducing drying time In this situation, the product developer (the experimenter) randomly assigned 10 test specimens to one formulation and 10 test specimens to the other formulation Then the paints were applied to the test specimens in random order until all 20 specimens were painted This is an example of a completely randomized experiment.

+ I(Qi-1,

When statistical signi cance is observed in a randomized experiment, the experimenter can be con dent in the conclusion that it was the difference in treatments that resulted in the difference in response That is, we can be con dent that a cause-and-effect relationship has been found Sometimes the objects to be used in the comparison are not assigned at random to the treatments For example, the September 1992 issue of Circulation (a medical journal published by the American Heart Association) reports a study linking high iron levels in the body with increased risk of heart attack The study, done in Finland, tracked 1931 men for ve years and showed a statistically signi cant effect of increasing iron levels on the incidence of heart attacks.

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