ams312final2010solutions

# ams312final2010solutions - AMS312.01 Final Exam Spring 2010...

This preview shows pages 1–3. Sign up to view the full content.

AMS312.01 Final Exam Spring, 2010 Name __________________________________ID ______________________Signature___________________ ______ Instruction: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please provide complete solutions for full credit. The exam goes from 2:15 - 4:45pm. Good luck! 1. A Gallup survey portrays U.S. entrepreneurs as "... the mavericks, dreamers, and loners whose rough edges and uncompromising need to do it their own way set them in sharp contrast to senior executives in major American corporations" (Wall Street Journal, May 1985). One of the many questions put to a sample of 100 entrepreneurs about their work habits, social activities, etc., concerned the origin of the car they personally drive most frequently. The responses are given in the following table. U.S. Europe Japan 45 46 9 Do these data provide evidence of a difference in the preference of entrepreneurs for domestic cars versus foreign cars? Test at α=0.05. Solution: This problem can be done in two ways using either (1) the test on one population proportion or (2) the Chi-square goodness-of-fit test with two categories. These two approaches are equivalent. (1) For the first approach, inference on one proportion, large sample, we have 100, 45 n x = = . Let p be the proportion of entrepreneurs with domestic cars, we have 45 ˆ 100 p = , and we are testing: 0 : 0.5 H p = versus : 0.5 a H p . The test statistics is: ( 29 0 ˆ 0.5 1 0.5 1 0.5 /100 p Z - = = - - Since 0 0.25 1 1.96 Z Z = < = , we can not reject the null hypothesis at the significance level of 0.05. (2) Alternatively, and equivalently, you can use the Chi-square goodness-of-fit test. The above table is readily reduced to the following two-category table: Domestic cars Foreign cars 1 45 x = 2 55 x = Let 1 2 , p p be the proportion of entrepreneurs with domestic or foreign cars respectively, we are testing: 0 1 2 : 0.5, 0.5 H p p = = versus 0 : a H H is not true. Hence we have 1 2 50, 50 e e = = . The test statistic is: 2 2 2 2 2 0 1,0.05, 0.025 1 ( ) 1 ( ) (1.96) 3.84 i i upper i i x e W Z e χ = - = = < = = Therefore we can not reject the null hypothesis at the significance level of 0.05. Of course you only need to show one of the two approaches above to get full credit. 2. ( 29 2 ~ , , 1,2,3 i i i X N i μ σ = and they are independent to each other, then (a) What is the distribution of 3 1 i i X = ? Prove your claim. (b) What is the distribution of 3 2 2 1 ( ) / i i i i X μ σ = - ? Prove your claim. Solution: (a) ( 29 [ ] ( 29 ( 29 ( 29 3 1 3 3 3 1 1 1 3 3 3 3 2 2 2 2 1 1 1 1 ( ) exp( ) exp( ) 1 1 exp( ) exp 2 2 i i i i i i i i i X X i i i i i i i i M t E exp X t E X t E X t M t t t t t μ σ μ σ = = = = = = = = = = = = = + = + Therefore we have shown that ( 29 ( 29 ( 29 3 3 3 2 1 1 1 ~ , i i i i i i X N μ σ = = = 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(b) Let i i i i X Z μ σ - = , then we have ( 29 2 2 2 2 1 exp exp exp 1 1 1 1 1 exp exp exp exp 2 2 i i i i i Z i i i i i i X i i i i i i i X M t E t E t X t t M t t t t t μ μ σ σ σ μ μ μ σ σ σ σ σ σ - - = = - - = = + =
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern