Economic Formula Sheet - Descriptive Statistics Page 3
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4Binomial Distribution
For a success probability of p and sample size n:
n!
(1 − p)
x
n−x
P (x successes) =
p
,
x!(n − x)!
and the discrete random variable x has mean
μ
= np
x
and variance
2
= np(1 − p)
σ
x
Sampling Distribution of the Sample Proportion
For large samples,
p(1 − p)
ˆ p ∼ N p,
n
Inference for Means
1. For large samples, a 100(1 − α)% CI for μ is:
(a)
σ
√
x ± z
α/2
n
2
when the population variance is known to be σ
;
(b)
s
√
x ± t
n−1,α/2
n
2
when the population variance is unknown but estimated by s
.
2. Differences in Means: Dependent Samples
− y
d
= x
,
1i
2i
i
then
n
d
i
i=1
d =
n
and
n
− d)
2
(d
i
i=1
s
=
d
n − 1
√
to form a confidence interval: d ± t
can be used with t
s
/
n.
d
n−1,α/2
n−1,α/2
3
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