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EXERCISE: Testing an Hypothesis
Regarding Pearson’s Data on Heights
From the evidence of Figure 1 of Lecture 1, which shows two regression lines ±tted
to Pearson’s data, there seems to have been a signi±cant increase in the heights of
adult males from one generation to the next in late Victorian England. The object
of this exercise is to test whether the increase is statistically signi±cant.
Let
x
denote the height of a father and
y
denote the height of his son. We may
assume that these heights are normally distributed with means of
E
(
x
)=
µ
x
and
E
(
y
µ
y
, with variances of
V
(
x
σ
2
x
and
V
(
y
σ
2
y
and with a covariance of
C
(
x, y
ρσ
x
σ
x
, which allows us to write
·
x
y
¸
∼
N
µ·
µ
x
µ
y
¸
,
·
σ
2
x
ρσ
x
σ
x
ρσ
x
σ
x
σ
2
y
¸¶
.
(1)
We shall also assume that
V
(
x
V
(
y
σ
2
, and we shall endeavour to test
the hypothesis that
E
(
x
E
(
y
µ
, given a sample of
N
observations contained
in the vectors
x
=[
x
1
,...,x
N
]
0
and
y
y
1
,...,y
N
]
0
. On that basis, there is
·
x
y
¸
∼
N
µ·
µ
x
ι
µ
y
ι
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This note was uploaded on 03/02/2012 for the course EC 3062 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.
 Spring '12
 D.S.G.Pollock

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