distribution is
given by
E
(
F
)
=
w
/(
w
−
2
)
;
it depends only on the second degrees of freedom
and is slightly larger than 1. The variance depends on both
v
and
w
. Table D
(at the end of the book) gives selected percentiles. For example, the 95 and 99th
percentiles of an
F
(5,8) distribution are given by 3.69 and 6.63, respectively.
Suppose
U
∼
χ
2
v
and
V
∼
χ
2
w
, with
U
and
V
independent. Then
F
=
U
/
v
V
/
w
has
an
F
distribution with
v
and
w
degrees of freedom.
A comment on statistical tables
: Most computer programs calculate cumu-
lative probabilities and percentiles (the
“
inverse
”
of the cumulative probabilities)
for a wide selection of distributions. For some programs (such as EXCEL) the
calculation of percentiles requires the speci
fi
cation of the upper tail area.
EXERCISES
2.1. Determine the 95 and 99th percentiles of
a. The normal distribution with mean 10 and
standard deviation 3;
b. The
t
distributions with 10 and 25 degrees
of freedom;
c. The chi-square distributions with 1, 4, and
10 degrees of freedom;
d. The
F
distributions with 2 and 10, and 4
and 10 degrees of freedom.
2.2. It is a fact that two distributions are the same
if (all) their percentiles are identical.
a. Convince yourself, by looking up several
percentiles, that the square of a standard
normal distribution is the same as a
chi-square distribution with one degree
of freedom. Determine the percentile of
the
χ
2
1
and the percentile of the square of
a standard normal distribution,
Z
2.3. For each of the four sets of data given below
(see Anscombe, 1973), plot
y
versus
x
. The
data are given in the
fi
le
anscombe
. Fit a
straight line model to each of the data sets
giving least squares estimates, ANOVA table,
and
R
2
. Compute the correlation coef
fi
cient
between
y
and
x
for each data set. Comment
on your results. Would a linear regression of
y
on
x
be appropriate in all cases? Discuss.
Set 1
Set 2
Set 3
Set 4
x
y
x
y
x
y
x
y
2
, and
show that they are the same. Use the fact
that
P
(
Z
2
≤
z
)
=
P
(
−
√
z
≤
Z
≤
√
z
)
.
Hence, for example, the 95th percentile
of
Z
2
is the same as the 97
.
5th percentile
of
Z
.
4
4.26
4
3.10
4
5.39
8
6.58
5
5.68
5
4.74
5
5.73
8
5.76
6
7.24
6
6.13
6
6.08
8
7.71
7
4.82
7
7.26
7
6.42
8
8.84
8
6.95
8
8.14
8
6.77
8
8.47
9
8.81
9
8.77
9
7.11
8
7.04
10
8.04
10
9.14
10
7.46
8
5.25
11
8.33
11
9.26
11
7.81
8
5.56
12
10.84
12
9.13
12
8.15
8
7.91
13
7.58
13
8.74
13
12.74
8
6.89
14
9.96
14
8.10
14
8.84
19
12.50

b. Convince yourself, by looking up several
percentiles, that the square of a
t
distribution with
v
degrees of freedom is
the same as the
F
(
1
,
v
)
distribution.
2.4. A car dealer is interested in modeling the
relationship between the weekly number of
cars sold and the daily average number of

Abraham
Abraham
˙
C02
November 8, 2004
0:36
Exercises
57
salespeople who work on the showroom
fl
oor during that week. The dealer believes
that the relationship between the two
variables can be described by a straight line.
The following data were supplied by the car
dealer:
Average. No.