lec7print

# lec7print - Normal distribution known variance Large sample...

This preview shows pages 1–7. Sign up to view the full content.

Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Confidence intervals Artin Armagan Sta. 113 Chapter 7 of Devore October 14, 2009 Artin Armagan Confidence intervals

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

View Full Document
Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Table of contents 1 Normal distribution known variance 2 Large sample CI, or CLT to the rescue 3 Small sample normal, thank Guinness 4 Confidence intervals on the spread or variance 5 Confidence bounds 6 Sample size computations Artin Armagan Confidence intervals
Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Uncertainty In the last lecture we learned about point estimates using the MLE. We also learned about uncertainty in the context of Bayesian methods and the posterior density. We now study within the likelihood framework how to think of uncertainty. This is the idea of a confidence interval and in statistics lingo it is the frequentist analog of the Bayesian credible interval. Artin Armagan Confidence intervals

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

View Full Document
Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Confidence interval of the mean If X 1 , ..., X n iid No( μ, σ 2 ) with then we know that Z = ¯ X μ σ/ n No(0 , 1) . This means that Pr ( 1 . 96 < Z < 1 . 96) = . 95 . Pr 1 . 96 < ¯ X μ σ/ n < 1 . 96 ! = . 95 . Pr 1 . 96 σ n < ¯ X μ < 1 . 96 σ n ! = . 95 . Pr 1 . 96 σ n ¯ X < μ < ¯ X + 1 . 96 σ n ! = . 95 . Pr 1 . 96 σ n + ¯ X > μ > ¯ X 1 . 96 σ n ! = . 95 . Pr ¯ X 1 . 96 σ n < μ < ¯ X + 1 . 96 σ n ! = . 95 . Artin Armagan Confidence intervals
Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations A random interval Consider the quantity Pr ¯ X 1 . 96 σ n < μ < ¯ X + 1 . 96 σ n ! = . 95 , ¯ X is random but μ is not it is fixed. The interpretation of the above equation is as a random interval = ¯ X 1 . 96 σ n , u = ¯ X + 1 . 96 σ n ! . The interval is centered at the sample mean and extends in either direction by 1 . 96 σ n . What a statistician would say is “the probability is . 95 that the random interval includes the true value μ. Artin Armagan Confidence intervals

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

View Full Document
Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Formal definition Definition Given x 1 , ..., x n iid No ( μ, σ 2 ) compute ¯ x. The 95% confidence interval for μ is parenleftbigg ¯ x
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