Prove the 2-tailed case of the One-Sample t-test, i.e., if X₁, X2,..., Xn is a random sample from a normal distribution with mean and unknown variance o2, then define 71 T= χ-μο S/√n with S2 (X-X). Hence for the test of hypothesis Ho: μ = po n-1 i=1 versus H₁ μμo, at the level of significance o, reject Ho iff either t ≤-ta/2(n-1) or t≥ ta/2(n-1).

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Prove the 2-tailed case of the One-Sample t-test, i.e., if X₁, X2,..., Xn is a random
sample from a normal distribution with mean and unknown variance o2, then define
71
1
X-μo
T =
S/√n
with S2
(X-X)². Hence for the test of hypothesis Ho: μ = μo
n-1
i=1
versus H₁ : μμo, at the level of significance a, reject Ho iff either t ≤-ta/2(n − 1)
or t≥ ta/2(n-1).
Transcribed Image Text:Prove the 2-tailed case of the One-Sample t-test, i.e., if X₁, X2,..., Xn is a random sample from a normal distribution with mean and unknown variance o2, then define 71 1 X-μo T = S/√n with S2 (X-X)². Hence for the test of hypothesis Ho: μ = μo n-1 i=1 versus H₁ : μμo, at the level of significance a, reject Ho iff either t ≤-ta/2(n − 1) or t≥ ta/2(n-1).
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