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Prove the 2-tailed case of the Two-Sample t-test, i.e., let X₁, X2,..., X be a random sample of size n from a normal distribution with mean #x and variance ² and let Y₁, Y₂,..., Ym be an independent random sample of size m from a normal distribution with mean µy and variance o2. Suppose that o2 is unknown. Let S and S be the corresponding sample variances and define the statistic X-Y-(HX-HY) T SpV m (n − 1)S² + (m − 1)S¾¸ Then for the test of hypothesis H₁ : µX = µy where S- = n+m-2 versus H₁ : µx # µy, reject Ho iff either t ≤-ta/2(v) or t≥ ta/2(v), wherev=n+m-2.

Prove the 2-tailed case of the Two-Sample t-test, i.e., let X₁, X2,..., X be a random sample of size n from a normal distribution with mean #x and variance ² and let Y₁, Y₂,..., Ym be an independent random sample of size m from a normal distribution with mean µy and variance o2. Suppose that o2 is unknown. Let S and S be the corresponding sample variances and define the statistic X-Y-(HX-HY) T SpV m (n − 1)S² + (m − 1)S¾¸ Then for the test of hypothesis H₁ : µX = µy where S- = n+m-2 versus H₁ : µx # µy, reject Ho iff either t ≤-ta/2(v) or t≥ ta/2(v), wherev=n+m-2.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Prove the 2-tailed case of the Two-Sample t-test, i.e., let X₁, X2,..., Xn be a random sample of size n from a normal distribution with mean x and variance o2 and let Y₁, Y2₂,..., Ym be an independent random sample of size m from a normal distribution with mean µy and variance o². Suppose that o² is unknown. Let S and S3 be the corresponding sample variances and define the statistic X-Y-(HX-HY) T= 1 SpV + n m (n − 1)S² + (m − 1) 5. Then for the test of hypothesis Ho: #x = HY where S = n+m-2 versus H₁ : μx μy, reject Ho iff either t ≤-ta/2(v) or t≥ ta/2(v), where v = n+m-2.
Transcribed Image Text:Prove the 2-tailed case of the Two-Sample t-test, i.e., let X₁, X2,..., Xn be a random sample of size n from a normal distribution with mean x and variance o2 and let Y₁, Y2₂,..., Ym be an independent random sample of size m from a normal distribution with mean µy and variance o². Suppose that o² is unknown. Let S and S3 be the corresponding sample variances and define the statistic X-Y-(HX-HY) T= 1 SpV + n m (n − 1)S² + (m − 1) 5. Then for the test of hypothesis Ho: #x = HY where S = n+m-2 versus H₁ : μx μy, reject Ho iff either t ≤-ta/2(v) or t≥ ta/2(v), where v = n+m-2.
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