STA 100: Confidence Interval Formulae
Normal CI for
μ
with known
σ
:
If
σ
is known
and
X
i
iid
∼
N
(
μ, σ
2
) for
i
= 1
, . . . , n
, then a
100(1

α
)% CI for
μ
is given by:
¯
x

z
1

α
2
·
σ
√
n
,
¯
x
+
z
1

α
2
·
σ
√
n
.
where
z
1

α
2
is the (1

α
2
)th percentile of a normal distribution. Depending on what level confidence
interval you want. . .
1.
90% CI:
α
= 0
.
10, (1

α
2
) = 0
.
950,
z
0
.
950
= 1
.
64;
2.
95% CI:
α
= 0
.
05, (1

α
2
) = 0
.
975,
z
0
.
975
= 1
.
96;
3.
99% CI:
α
= 0
.
01, (1

α
2
) = 0
.
995,
z
0
.
995
= 2
.
58.
Normal CI for
μ
with unknown
σ
If
X
i
iid
∼
N
(
μ, σ
2
) for
i
= 1
, . . . , n
, and
σ
is unknown
then
a 100(1

α
)% CI for
μ
is given by:
¯
x

t
n

1
,
1

α
2
·
s
√
n
,
¯
x
+
t
n

1
,
1

α
2
·
s
√
n
.
where
s
is the sample standard deviation and
t
n

1
,
1

α
2
is the (1

α
2
)th percentile of a
t
n

1
distri
bution i.e., the (1

α
2
)th percentile of a
t

distribution with
n

1 degrees of freedom.
To find what
t
n

1
,
1

α
2
is in
R
, type:
qt(1alpha/2,df=n1)
where you replace
n
and
alpha
with the appropriate numbers!
1.
90% CI:
α
= 0
.
10, (1

α
2
) = 0
.
950,
t
n

1
,
0
.
950
=
qt(0.950,df=n1)
;
2.
95% CI:
α
= 0
.
05, (1

α
2
) = 0
.
975,
t
n

1
,
0
.
975
=
qt(0.975,df=n1)
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