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07C_woEx_F10

# 07C_woEx_F10 - The Hong Kong University of Science...

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The Hong Kong University of Science & Technology ISMT 111: Business Statistics Tutorial Set 7 Tests of Hypotheses for Means and Proportions De fi nition Null Hypothesis ( H 0 ) A hypothesis we assume to be TRUE, e.g. H 0 : μ = μ 0 Alternative Hypothesis ( H a ) A hypothesis we wish to support, e.g. H a : μ 6 = μ 0 Test Statistics ( z obs or t obs ) A value used to make the decision, e.g. z obs = X μ 0 σ / n Critical Value ( c ) A value used to separate the rejection and non-rejection regions α The Level of Signi fi cance 1- β The Power of the Test p -value P ( z > | z obs | ) ; P ( t > | t obs | ) ; the smallest value of α to reject H 0 1

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Type I and Type II Error H 0 is TRUE H 0 is FALSE X Type II Error Do NOT Reject H 0 P ( Not Reject H 0 | H 0 is True ) β = P ( Committing Type II Error ) = 1 α ( Con fi dence Coe cient ) = P ( Not Reject H 0 | H 0 is False ) Type I Error X Reject H 0 P ( Committing Type I Error ) P ( Reject H 0 | H 0 is False ) = P ( Reject H 0 | H 0 is True ) = 1 β ( Power of the Test ) = α ( Signi fi cance Level ) NOTE For fi xed n , α = β or α = β n = α = β p -value < α = Reject H 0 at α level p -value > α = Do not reject H 0 at α level 2
Hypothesis Testing of a Population Mean μ Null Hypothesis H 0 : μ = μ 0 Alternative Hypothesis Test Statistics Rejection Region / Decision Large Sample H a : μ < μ 0 z obs < z α (i) Population normal; σ known H a : μ > μ 0 z obs = X μ 0 σ / n z obs > z α (ii) Population not normal; σ unknown; large sample ( n 30) H a : μ 6 = μ 0 z obs = X μ 0 s/ n z obs < z α 2 or z obs > z α 2 Small Sample ( n < 30) H a : μ < μ 0 t obs < t α Population normal with unknown σ H a : μ > μ 0 t obs = X μ 0 s/ n t obs > t α H a : μ 6 = μ 0 ( d.f. = n 1) t obs < t α 2 or t obs > t α 2 3

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Procedures of Hypothesis Testing An Example
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07C_woEx_F10 - The Hong Kong University of Science...

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