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Hypothesis testing Another judgment method of sampling data

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Hypothesis Null hypothesis If a population having a distribution function F θ, where θ is unknown, and suppose we want to test a specific hypothesis about θ such as θ=1. We denote this hypothesis H 0 and call it the null hypothesis The word “ null ” means we can not assert confidentially this hypothesis approaches to be true until it is tested by the observed data set. Simple hypothesis: Completely specify the population distribution when null hypothesis is true. E.g., H 0 : θ=1 Composite hypothesis Don ’ t specify the population distribution when null hypothesis is true. E.g., H 0 : θ ≧ 1

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Critical region The criteria for rejecting or accepting the null hypothesis Specify the specific range or the upper/lower bound of a certain statistics. Denote the critical region, C. If the resultant sample make the calculated statistics lied in C, then we will reject H 0, otherwise, accept H 0.

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Type I error and type II error Type I The test incorrectly calls for rejecting H 0 when it is indeed correct. Prob. (reject H 0 /H 0 is true) Type II The test calls for accepting H 0 when it is false. Prob. (accept H 0 /H 0 is false)

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Significance level H 0 should only be rejected if the resultant data are very unlikely when H 0 is true. The significance level α, commonly chosen value being.01,.05, is the error probability that H 0 is rejected when H 0 is indeed true. The α, in other words, is the maximum type I error that the tester could endure. If the error probability of rejecting H 0 is greater α, then the researcher had better to accept H 0.

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“ Far away ” from the value of H 0 H 0 : θ ≦ ω (θ is the unknown parameter and ω is a particular value) If the estimator of θ, d(X), is far away from the specified region ω, then we need to consider the rejection of H0 and moreover to justify whether the appropriate significance level α is.

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Alternative hypothesis We are supposed to be interested in testing the null hypothesis, H 0 :θ=θ 0, against the alternative hypothesis, H1:θ≠θ 0, where θ 0 is some specified constant. H 0 (stated the current situation) in contrast to H 1 (stated the experimental/desired situation)

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Testing the mean of a normal population — case of known variance, H 0 : μ=μ 0 vs. H 1 : μ≠μ 0 Critical region ∵∴ ∴

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The test statistics distribution when H 0 is true

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P-value of a hypothesis test If the observed value of test statistics, ν, lies within the critical region, then the occurring probability of such a worse ν, we call it the p-value, must be less than or equal to α (the significance level). In other words, if the p-value is smaller than the chosen significance α, then we will reject H 0.

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Operating characteristic (OC) curve The OC curve: the probability of accepting “ H 0: μ = μ 0 “ under the true mean is equal to μ

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Operating characteristic (OC) curve (cont.) If μ = μ 0, then d=0 & β( μ )= α =0.95. If μ is very large than μ 0, then β(μ) will reduce to zero.

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The power of test The function 1-β(μ) is called the power-function of the test. The power of the test is equal to the probability of rejection when μ is the true value. The more OC β(μ) is, the larger probability of accepting H 0 is. The more power of testing is, the larger probability of rejection H 0 is. The type II error=prob{accept H 0 /H 0 is false}=β(μ=μ 1 ), whereμ 1 ≠μ 0 Determine the necessary sample size n under the endurable type II error β, ref. pp.298-299

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One-side tests, H 0 :μ= ( ≦ ) μ 0 vs.H 1 :μ ＞ μ 0

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The t-test – the case of unknown variance, H 0 : μ=μ 0 vs. H 1 : μ≠μ 0 Reject when it is large

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The t-test under the unknown variance

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Testing the equality of means of two normal populations — case of known variances, H 0 : μ x =μ y vs. H 1 :μ x ≠μ y In the same way, H 0 :μ x ≦ μ y vs. H 1 :μ x ＞ μ y

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Testing the equality of means of two normal populations — case of unknown but equal variances, H 0 : μ x =μ y vs. H 1 :μ x ≠μ y Set ∴ In the same way,

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Testing the equality of means of two normal populations — case of unknown and unequal variances, H 0 : μ x =μ y vs. H 1 :μ x ≠μ y Even under the case of unknown and unequal variances

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The paired t-test Set the new random variable W i, and rearrange W i =X i -Y i, i=1,2, … n H 0 : μ w =0 vs. H 1 : μ w ≠0 Compute the average of W and use the t- test for hypothesis testing

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Hypothesis testing of the variance of a normal population ∵

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Hypothesis testing for the equality of variances of two normal populations

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Hypothesis tests in Bernoulli populations

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Approximate p testing

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Homework #1 Problems 10,26,28,31,43,55

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