Economic Formula Sheet - Descriptive Statistics Page 4
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1
2
3
43. Differences in Means: Independent Samples
(variances unknown and not assumed equal):
2
2
s
s
1
2
s
=
+
−x
x
1
2
n
n
1
2
can be used with t
and
ν,α/2
2
2
2
[(s
/n
) + (s
/n
)]
1
2
1
2
ν =
,
2
2
2
− 1) + (s
2
− 1)
(s
/n
)
/(n
/n
)
/(n
1
1
2
2
1
2
− 1 and n
− 1.
or often degrees of freedom ν approximated by the smaller of n
1
2
Inference for Proportions
1. For large samples, a 100(1 − α)% CI for p is:
ˆ p (1 − ˆ p )
ˆ p ± z
.
α/2
n
(Or replace ˆ p = x/n by ˜ p = (x + 2)/(n + 4) when α = 1%, 5%, or 10%.)
2. Differences in Proportions
− ˆ p
ˆ p
1
2
has standard deviation:
(1 − ˆ p
(1 − ˆ p
ˆ p
)
ˆ p
)
1
1
2
2
+
n
n
1
2
which can be used with z
to form a confidence interval. (Or add 1 success and 1 failure to each sample
α/2
when α = 1%, 5%, or 10%.)
3. Testing the Hypothesis of Equal Proportions
For this test, use the pooled estimate of the common value of p
and p
:
1
2
X
+ X
1
2
ˆ p
=
pool
n
+ n
1
2
to form
1
1
(1 − ˆ p
SE
=
ˆ p
)
+
Dp
pool
pool
n
n
1
2
4
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