# Freedom of degrees 1 with on distributi the using

• Notes
• 27

This preview shows pages 15–24. Sign up to view the full content.

freedom of degrees 1 with on distributi the using determined be will values critical but rules decision define to used be to statistic test the is Similarly for CI )% - (100 the is 1 , 2 / n- t n s X t n s t X n μ μ α α - = ± -

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

16 Final exam June 2005 Q3(b) l Assume that an investor has a portfolio of investments l From past evidence, annual returns can be treated as approximately normal l In a small sample of 9 investments, the sample mean=15% & (sample) standard deviation=11% l In question were not told but you needed to realize l CLT is inappropriate, but sampling from a normal population, so not needed
Final exam June 2005 Q3(b)... l Consider the following questions l (i) Assume population standard deviation to be 12%. Suppose CI has a lower limit of 8.42%. What confidence level was used? l (ii) If use a higher confidence level what happens to the width of the CI? l (iii) Assume the population standard deviation is unknown. Find the 95% CI for the population mean return l (iv) Interpret the numerical result in (iii). 17

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

18 Final exam June 2005 Q3(b)… interval wider confidence of level in increase se (ii)Decrea CI 90% a used hence & % 10 05 . ) 645 . 1 ( 645 . 1 4 / ) 42 . 8 15 ( 3 12 - 15 8.42 or n be will bound lower & n is CI )100% - (1 then 12 If (i) ) 1 , 0 ( ~ and /n) , ( then ) , ( be returns let & 11 , 15 , 9 Know 2 / 2 / 2 / 2 / 2 2 = = = - = = - ± = - = = = = α α σ σ α σ σ μ σ μ σ μ α α α α Z P z z z X z X N n X Z ~N X X~N s x n
19 Final exam June 2005 Q3(b)… return mean population the include would manner this in d constructe intervals the of 95% that expect would you returns of population this from drawn 9 size of samples repeated (iv)In ) 46 . 23 , 54 . 7 ( or 3 11 .306 2 15 n by given CI 95% & ~ (iii) of use unknown Assuming 1 , 2 / 1 ± ± - - - s t X t n s X t n n α μ σ

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

20 Final exam June 2005 Q3(b)… l How would you answer (iii) if you were not willing to assume normality for the underlying population?
21 Inferences about other parameters? l Have concentrated on inference for the population mean l Concepts & procedures can be applied to testing & creating CI’s for many other situations, including l Population proportion l Regression parameters l Difference in means l Population variance (first considered now, other 3 will be treated later in course)

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

22 Estimating the population proportion l Recall binomial rv X l Number of successes in n trials l Know that E( X )= np & Var( X )= np(1-p) l Now consider the rv X / n l This is simply the sample proportion of successes l It is an unbiased estimator of the population proportion l Why? l It has a variance of p(1-p)/n l Why?
23 Estimating the population proportion… l Also recall that for large n, X is approximately normal l Thus the sample proportion is also approximately normal for large n using the CLT l Approximate sampling distribution follows immediately l As do confidence intervals & hypothesis tests ) 1 , 0 ( ~ / ) 1 ( ˆ and ) 1 ( , ˆ then ˆ Let N n p p p P Z n p p p ~N P n X P - - = - =

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

This is the end of the preview. Sign up to access the rest of the document.
• Three '11
• DenzilGFiebig

{[ 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