Assignment 6 sol

# Assignment 6 sol - The University of Hong Kong Department...

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The University of Hong Kong Department of Statistics and Actuarial Science STAT0302 Business Statistics Semester 2 2008/2009 Assignment 6 Suggested Solution Q1. CHI-SQUARED TEST OF INDEPENDENCE: H F L P Total A 5 10 11 14 40 O 14 16 16 12 58 U 10 7 4 4 25 Total 29 33 31 30 123 0 H : Appearance is INDEPENENT of academic performance. A H : The two factors are NOT independent. Test statistic ( Under ): 0 H 2 ) 1 )( 1 ( 2 2 ~ } 1 . { = ∑∑ c r ij j i ij t s o n χ where and are the row and column totals for each observation . i s j t 97224 . 8 2 = Tables : Look up – tables: df = (r-1)(c-1)=2x3=6, since r=3 and c=4 2 α = 0.05 one-tailed which gives 12.59 as the critical value. Conclusion: Since the test statistic value 8.97224 is clearly < the critical value = 12.59 we DO NOT REJECT at the α = 5% level of significance. Hence, there is no relationship between a student’s academic performance and their appearance. 2 6 0 H

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Q2. USE TWO METHODS *********** Let , and be the proportion of voters favouring the legislation of low, average and high incomes respectively. A p B p C p 7 . 0 150 105 ˆ = = A p , 6 . 0 160 96 ˆ = = B p and 42 . 0 100 42 ˆ = = C p 6484 . 0 310 201 ˆ = = AB p , 588 . 0 250 147 ˆ = = AC p and 5308 . 0 260 138 ˆ = = BC p i) Hypothesis : 0 : 0 : 0 = = B A B A A B A B A p p p p H p p p p H Test statistic: ) 1 , 0 ( ~ ) 1 1 )( ˆ 1 ( ˆ ) ( ) ˆ ˆ ( N n n p p p p p p Z B A AB AB B A B A + = 8428 . 1 ) 160 1 150 1 )( 3516 . 0 )( 6484 . 0 ( 0 6 . 0 7 . 0 = + = Tables: Look up 0.475 in z – tables which gives 1.96 as the critical value. Conclusion: Since the test statistic value Z = 1.8428 < the critical value z = 1.96, so we DO NOT REJECT at the α = 5% level of significance. Hence, among low incomes and average incomes classes, the proportions of voters favouring the legislation are the same. 0 H ii) Hypothesis: 0 : 0 : 0 = = C A C A A C A C A p p p p H p p p p H
Test statistic: ) 1 , 0 ( ~ ) 1 1 )( ˆ 1 ( ˆ ) ( ) ˆ ˆ ( N n n p p p p p p Z C A AC AC C A C A + = 4065 . 4 ) 100 1 150 1 )( 412 . 0 )( 588 . 0 ( 0 42 . 0 7 . 0 = + = Tables: Look up 0.475 in z – tables which gives 1.96 as the critical value. Conclusion: Since the test statistic value Z = 4.4065 > the critical value z = 1.96, so we REJECT at the α = 5% level of significance. Hence, among low incomes and high incomes classes, the proportions of voters favouring the legislation are not the same. 0

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Assignment 6 sol - The University of Hong Kong Department...

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