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Quiz

May 12, 2016

0.1 Quiz

Suppose a sample of size 3 is taken from a uniform distribution over the interval [0 , ], for the purpose oftesting

H0 : = 1

versus (1)

H1 :

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In [59]: y

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The pdf ofW=Y1+ Y2+ Y3 is given by

fW(w; ) =

w2

23 0 w <

w2 + 6w 32

23 w

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The probability of committing type II error is given by

=

1.5

0.6

fW(w; 0.5) dw= 0.716

In [3]: library(png)

img

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So, we have

Var(Yi) = 2

12

The expected value ofZis given by

=3

i=1

2

(5)

= 3

2 (6)

and The variance is

Var(W) =

3i=1

2

12 (7)

= 2

4 (8)

We can compute as follows.

z = y w

w

= 0.6 1.5

0.5 = 1.8

= 0.0359

In [13]: pnorm(0.6,3/2,1/2)

Out[13]:

0.0359303191129258

If = 0.5 then we have W N(3/4, 1/16)

z = y w

w(9)

= 0.6 0.75

0.25 = 0.6 (10)

= 1 0.276 = 0.7257 (11)

In [1]: 1-pnorm(0.6,3/4,1/4)

Out[1]:

0.725746882249926

In [4]: img1

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