{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

handout4_1

# handout4_1 - Econ 139 Introduction to Econometrics Andrew...

This preview shows pages 1–10. Sign up to view the full content.

Econ 139: Introduction to Econometrics Andrew Sweeting 1 Department of Economics Duke University Spring 2011 Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 1 / 47

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

View Full Document
Con°dence Intervals and Hypothesis Testing (Chapter 3) We usually estimate something with a speci°c question/hypothesis in mind. e.g., did most Duke faculty vote for Obama? suppose we interview a sample of faculty and °nd that 45% of the sample voted for Obama what can we conclude? is this consistent with a hypothesis that most faculty supported Obama? to do this we will use our knowledge of the estimator±s distribution to construct con°dence intervals and perform hypothesis tests . Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 2 / 47
Con°dence Intervals A con°dence interval de°nes an interval around the estimate in which we expect the population parameter to lie with some probability. E.g., Pr ( ¯ X ° error < μ X < ¯ X + error ) = 0 . 95 0.95 is the standard probability used and re²ects the fact that we want to be conservative (i.e., wide intervals around our estimate). Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 3 / 47

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

View Full Document
Constructing a Con°dence Interval Let±s begin with a closely related exercise. Find an interval that will contain a random draw from a standard normal (which has mean zero) distribution 95% of the time From our previous analysis and the standard normal tables we know Pr ( ° 1 . 96 < Z < 1 . 96 ) = 0 . 95 3 1 0 -1 -3 0.95 0.025 0.025 1.96 -1.96 There is a 95% chance that the random variable Z will fall in the interval ( ° 1 . 96 , 1 . 96 ) Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 4 / 47
Constructing a Con°dence Interval Now let±s construct a 95% con°dence interval for the mean μ X based on the sample mean estimate ¯ X From the CLT we know that ¯ X ° μ X σ X p n is (approximately) N ( 0 , 1 ) . Therefore Pr ° ° 1 . 96 < ¯ X ° μ X σ X p n < 1 . 96 ± = 0 . 95 as above. Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 5 / 47

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

View Full Document
Constructing a Con°dence Interval Pr ° 1 . 96 < ¯ X ° μ X σ X p n < 1 . 96 ! = 0 . 95 Rearranging, Pr ° ¯ X ° 1 . 96 ± σ X p n < μ X < ¯ X + 1 . 96 ± σ X p n ± = 0 . 95 . Thus, a 95% con°dence interval for μ X is ¯ X ² 1 . 96 ± σ X p n = ² ¯ X ° 1 . 96 ± σ X p n , ¯ X + 1 . 96 ± σ X p n ³ Notice that the interval gets tighter as n °! Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 6 / 47
Example we test 100 cars and °nd the sample mean MPG is 20.5 magic tells us that σ 2 = 16 construct the 95% con°dence interval: Pr ° ¯ X ° 1 . 96 ± σ X p n < μ X < ¯ X + 1 . 96 ± σ X p n ± = 0 . 95 . Pr ° 20 . 5 ° 1 . 96 ± 4 10 < μ X < 20 . 5 + 1 . 96 ± 4 10 ± = 0 . 95 ) ² 20 . 5 ° 1 . 96 ± 4 10 , 20 . 5 + 1 . 96 ± 4 10 ³ = [ 19 . 7 , 21 . 3 ] Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 7 / 47

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

View Full Document
Interpretation ¯ X ² 1 . 96 ± σ X p n = ² ¯ X ° 1 . 96 ± σ X p n , ¯ X + 1 . 96 ± σ X p n ³ Which statement about the interval above is correct? 1 ³There is a 95% chance that the population mean μ X is contained in the interval.´ 2 ³There is a 95% chance that the interval will contain the population mean μ X . ´ Econ 139 Handout 4 (Duke) Hypothesis Testing Spring 2011 8 / 47
Interpretation ¯ X ² 1 . 96 ± σ X p n = ² ¯ X ° 1 . 96 ± σ X p n , ¯ X + 1 . 96 ± σ X p n ³

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 47

handout4_1 - Econ 139 Introduction to Econometrics Andrew...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online