Week 11 Practice Problems
1.
A sample of 200 people are each asked to produce a “random digit” between 1 and 5.
The data are
value
1
2
3
4
5
frequency
35
42
48
37
38
Does a uniform model with equal probability for each digit fit these data?
Step 1: Suppose the uniform model is appropriate
Step 2: The multinomial likelihood is
35
42
48
37
38
12345
1 2
3
4
5
(, , , , )
,
1
Lc
θθθθθ
θθθθθ θθθθθ
=+
+
+
+
=
With no restrictions, we have
ˆˆ
ˆ
35 / 200,
42 / 200,
48 / 200,
37 / 200,
38 / 200
θθ
θ
=====
Assuming the uniform model we have
0.2,
1,.
..,5
j
j
=
=
Step 3: The (likelihood ratio) discrepancy measure is
1234
ˆˆˆˆˆ
2ln[ (0.2,0.2,0.2,0.2,0.2) / ( ,
,
,
,
)]
ˆˆˆˆ
2[35ln(0.2 /
) 42ln(0.2 /
)
48ln(0.2 /
) 37ln(0.2 /
) 38ln(0.2 /
)]
2.59
dL
L
5
ˆ
θθθ
=−
+
+
+
+
=
Step 4: The p-value is
2
4
(
)
(
2.59)
0.10
PD d
P
χ
≥≈
≥
>
Step 5: There is no evidence that
j
is different from 0.2
There is no evidence against the fit of the uniform model.
2.
In an ESP experiment, four cards labeled A,B,C,D are placed face down on a table
and a subject is asked to peer through the cards in his or her mind and identify which
card is which.
a)
Show that if a subject guesses which card is which, then the probability
function for the correct number of matches
Y
is
y
0
1
2
4
(
PY
y
=
)
9/24
8/24
6/24
1/24