Chapter 10

# Δ μ μ n z α z β 2 σ 2 δ 2 sample size for t

• Notes
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( δ = μ - μ 0 ) n = (z α + z β ) 2 σ 2 / δ 2

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Sample Size for t test for One Mean This is a two step process: - Identify ∆ = | δ|/σ = | μ - μ 0 |/σ - Use the table to obtain the sample size using ∆, α, β
Example We wish to detect a difference equal to 6/10 of a standard deviation (.6 σ ) when completing a test with H 1 : μ > 25 for alpha and beta of .05.

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Determining Sample Sizes Needed for Tests with Two Means Just as in the one sample tests, there are times we want to know what sample size is needed to reach a certain level of alpha and beta. In other words, what sample size is needed to keep the probability of type I and type II errors to a certain level? 31
Sample Size for z test for Two Means One tailed alternative hypothesis: n = (z α + z β ) 2 1 2 + σ 2 2 ) / δ 2 where δ is the difference between the true difference in means (d= μ 1 - μ 2 ) and the difference in the means in our hypotheses (d 0 = μ 1 - μ 2 ) Two tailed alternative hypothesis: n = (z α/2 + z β ) 2 1 2 + σ 2 2 ) / δ 2 Always round up to closest integer.

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Sample Size for Pooled t test for Two Means This is a two step process: - Identify ∆ = | δ|/σ = |(μ 1 - μ 2 29- d 0 | - Use Table A.9 to obtain the sample size using ∆, α, β
Example In comparing the performance of two catalysts on the effect of a reaction yield, a two sample t-test is conducted with alpha = 0.05. The variances are considered to be the same for the two catalysts. How large of a sample for each catalyst is needed to test the hypothesis H0: μ 1 = μ 2 vs. H1: μ 1 does not equal μ 2 if it is essential to detect a difference of 0.8 σ between the catalysts with probability 0.9? Beta = 1 – power = 1 - .9 = .10 | μ 1 - μ 2 - d 0 | /σ =
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• Spring '07
• MQLemons
• Statistics, Variance, Null hypothesis, Statistical hypothesis testing, Type I and type II errors, Student's t-test

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