PMid2 - AMS312 Spring 2010 Practice Midterm #2 &...

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Unformatted text preview: AMS312 Spring 2010 Practice Midterm #2 & Solution (***Note: the real midterm will have only 3 questions. Here I provided more to help you review.) 1. Arctic and Alpine Research investigated the relationship between the mean daily air temperature and the cocoon temperature of woolybear caterpillar’s of the High Arctic. (a) According to the data, can you conclude, at the significance level of 0.05, that the caterpillar’s body temperature is higher than the outside air temperature? (b) What assumptions are necessary for the above test? Temperature (ºC) Day Air Cocoon 1 10 15 2 9 14 3 2 7 4 3 6 5 5 10 Solution: By taking the paired differences (Diff) between the cocoon and the air temperatures for each day sampled, this problem reduce to a one-sample t-test on Diff. Temperature (ºC) Day Air Cocoon Diff 1 10 15 5 2 9 14 5 3 2 7 5 4 3 6 3 5 5 10 5 (a). Sample statistics: n = 5, , 6 . 4 = x s = 0.9. Hypotheses: H : μ = 0 versus Ha: μ > 0. Test statistic (observed): 5 . 11 5 / 9 . 6 . 4 / ≈- =- = n s x t Since , 13 . 2 5 . 11 05 . , 4 = ≈ t t we reject H 0 in favor of Ha at the 0.05 significance level. That is, we conclude, at the significance level of 0.05, that the caterpillar’s body temperature is higher than the outside air temperature. (b). The assumption is that the population distribution of “Diff” is normal. 2. Suppose a random sample of size n is drawn from a normal population with mean μ and variance σ 2 , both parameters unknown. Please a. Find the maximum likelihood estimators of μ and σ 2 . b. Find the method of moment estimators of μ and σ 2 . Solution: 1 (a) 2 2 2 2 2 1 1 ( ) 1 2 ( , ) ( ; , ) ( ) n n i i i i x L f x e π μ σ σ μ σ μ σ = =-- = = ∏ ∏ Ln L= 2 2 2 2 2 1 1 1 2 ( ) ( ) ln( 2 ) ln( 2 ) 2 2 ln i i n n i i x x μ μ πσ π σ σ σ = =------- = ∑ ∑ 1 1 1 2 2 2 2( ) 2 ln ( ( 1)) n n i i n i i i i x n x x L μ μ μ σ σ σ μ = = =--- ∂ =-- = = ∂ ∑ ∑ ∑ 2 2 2 2 2 1 2 4 4 4 2 1 1 ( ) ( ) ( ) 1 2 2 2 2 ln ( ) n n n i i i i i i x n x x L μ σ μ μ σ σ σ σ σ σ = = =------ ∂ = + = = ∂ ∑ ∑ ∑ µ 1 ln 0 n i i x n L x μ μ = ∂ = ⇒ = ∂ ∑ = (1) ¶ 2 2 1 2 ( ) ln n i i x n L μ σ σ...
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This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.

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PMid2 - AMS312 Spring 2010 Practice Midterm #2 &...

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