The critical value
,
t
is that value of t for which the area to its right under the student’s t
distribution is equal to
, given degrees of freedom
.
Graphically,
For example, suppose there are 20 degrees of
freedom and we want to find the value of t
such that the area to the right of it is 0.05.
Locate 20 in the first column of the table and
read across to the entry located in column
t
0 05
.
to get 1.725.
Now suppose we want to find the value of t
such that the area to the
left
of it is 0.05,
given
20
.
Because the t distribution is
symmetric about zero, this value of t is
t
0 05
1725
.
,20
.
.
27
USING THE CHISQUARED TABLE OF CRITICAL VALUES
The Chisquared statistic is nonnegative and its distribution is positively skewed. As with the
student t distribution, the shape of the chisquared distribution depends on the number of
degrees of freedom.
The critical value of
,
2
is that value of
2
for which the area to its right under the Chi
squared distribution is equal to
, given degrees of freedom
.
Graphically,
For example, suppose there are 10 degrees of
freedom and we want to find the value of
2
such that the area to its right is 0.05.
Locate
10 in the first column (degrees of freedom)
and read across to the entry located in
column 0.05 to get 18.3070.
Now suppose we want to find the value of
2
such that the area to the
left
of it is 0.05,
given
10.
Because,
2
is a nonnegative
random variable, we must find that value of
2
such that the area to its right 0.95.
Locate 10 in the first column and read across
to the entry located in column 0.95 to get
3.94030.
28
THE GREEK ALPHABET
Letters
Names
English
Equivalent
Letters
Names
English
Equivalent
A
Alpha
A
N
Nu
n
B
Beta
B
Xi
x
Gamma
G
O o
Omicron
o
Delta
D
Pi
p
E
Epsilon
E
P
Rho
r
Z
Zeta
Z
Sigma
s
H
Eta

T
Tau
t
Theta

Y
Upsilon
u or y
I
Iota
I
Phi

K
Kappa
K
X
Chi

Lambda
L
Psi

M
Mu
M
Omega

29
USEFUL FORMULAE
Sample Mean:
n
i
i
X
n
X
1
1
Sample Variance:
2
1
2
)
(
1
1
X
X
n
s
n
i
i
Population Variance:
2
1
2
)
(
1
N
i
i
X
N
Population Mean:
N
i
i
X
N
1
1
Additive Law of Probability:
P A
B
P A
P B
P A
B
(
)
(
)
(
)
(
)
Multiplicative Law of Probability:
P A
B
P A B P B
P B A P A
(
)
(
)
(
)
(
)
(
)
Binomial Distribution:
x
n
x
x
n
q
p
C
x
X
P
)
(
E(X) = np;
Var(X) = npq
Standardising transformations:
)
(
X
Z
,
n
X
Z
/
)
(
,
n
s
X
t
/
)
(
,
2
2
2
1
s
n
Confidence intervals for
:
n
z
X
n
z
X
2
2
,
known
n
s
t
X
n
s
t
X
n
n
1
,
2
1
,
2
,
unknown
Confidence interval for p:
n
q
p
z
p
p
n
q
p
z
p
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
2
,
where
p
q
ˆ
1
ˆ
Confidence interval for
σ
2
:
2
1
,
2
/
1
2
2
2
1
,
2
/
2
)
1
(
)
1
(
n
n
s
n
s
n
Goodness of Fit Test:
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 '08
 julia
 Normal Distribution, Standard Deviation, tutorial exercises