Palgrave Handbook of Econometrics: Applied Econometrics

John DiNardo 125

First, let’s illustrate the problem. In experiment A, the question is: “How often
would we expect to see 9 black balls out of 12 balls under the null hypothesis?”:

P(̂μ≥

3

4

|H 0 ) ≡ P(X≥ 9 |H 0 )

=

∑^12

x= 9

(

12

x

)

μx( 1 −μ)^12 −x

=

(

12

9

)

1

2

9

( 1 −

1

2

)^3 +

(

12

10

)

1

2

10

( 1 −

1

2

)^2 +...

=

220 + 66 + 12 + 1

212

=

299

212

= 0.073.

In experiment B, the question is: “Under the null hypothesis, what is the prob-
ability of drawing 9 or more black balls before drawing a third red ball?” Letr= 3
be the pre-specified number of red balls to be drawn before the experiment is to be
stopped. Letxindex the number of black balls drawn, and letn=x+r.
This is a straightforward application of the negative binomial distribution
where:

P(X≥ 9 |H 0 ) =

∑∞

x= 9

(

r+x− 1

r− 1

)

μx( 1 −μ)r

=

∑∞

x= 9

(

x+ 2

2

)

μx( 1 −μ)r.

It is very helpful to observe in doing the calculation that:

∑∞

x=j

(

x+ 2

2

)

(

1

2

)x=

8 + 5 j+j^2

2 j

.

We can then write:

=

∑∞

x= 9

(

x+ 2

2

)

μx( 1 −μ)^3

=

(

1

2

) 3

8 + 5 ( 9 )+ 92

29

=

1

8

(

134

512

)

= 0.0327.