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Unformatted text preview: You may be wondering why we claim that “we reject H 0 in a two-tailed test with significance level α ” and “the (1 − α ) × 100% confidence interval does not contain µ0 (or D0 ) ” are equivalent statements. Here is an algebraic proof that the two statements are the same: First, we start with the statement, “we reject H 0 in a two-tailed test with significance level α . ” For a z-test, this means that zobs > zα 2 . Substituting in the equation for zobs , we have: x − µ0 > zα 2 σn
Since σ n is always positive, we have the following two possible scenarios, depending on whether x − µ0 is positive or negative: − ( x − µ0 ) x − µ0 > zα 2 or > zα 2 σn σn Rearranging these two inequalities, we have:
x − µ0 > zα 2 σ n or − ( x − µ0 ) > zα 2 σ
n − µ0 > − x + zα 2 σ
n or − x + µ0 > zα 2 σ
n n n The last statement says that µ0 is outside the (1 − α ) × 100% confidence interval constructed around x . Therefore we have proven the original claim for a z-test, and the proof for a t-test is similar. µ0 < x − zα 2 σ or µ0 > x + zα 2 σ ...
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- Spring '10