# Sampling Distributions Of Differences And Sums Of Statistics, - HP 48gII User Manual

Graphing calculator.

is the probability of success, then the mean value, or expectation, of X is E[X]
= p, and its variance is Var[X] = p(1-p).
If an experiment involving X is repeated n times, and k successful outcomes
are recorded, then an estimate of p is given by p'= k/n, while the standard
error of p' is σ
= √(p⋅(1-p)/n) . In practice, the sample estimate for p, i.e., p'
p'
replaces p in the standard error formula.
For a large sample size, n>30, and n⋅p > 5 and n⋅(1-p)>5, the sampling
distribution is very nearly normal. Therefore, the 100(1-α) % central two-sided
confidence interval for the population mean p is (p'+z
For a small sample (n<30), the interval can be estimated as (p'-t
⋅σ
).
α
1,
/2
p'
Sampling distribution of differences and sums of statistics
Let S
and S
be independent statistics from two populations based on
1
2
samples of sizes n
and n
1
standard errors of the sampling distributions of those statistics be µ
and σ
and σ
, respectively.
S1
S2
, have a sampling distribution with mean µ
two populations, S
-S
1
2
µ
, and standard error σ
S2
has a mean µ
T
+T
1
2
S1+S2
σ
2
1/2
)
.
S2
Estimators for the mean and standard deviation of the difference and sum of
the statistics S
and S
are given by:
1
2
ˆ
µ
=
S
±
S
1
2
In these expressions, X
samples taken from the two populations, and σ
of the populations of the statistics S
taken.
, respectively. Also, let the respective means and
2
The differences between the statistics from the
2
+ σ
2
1/2
= (σ
)
S1
S2
S1
S2
= µ
, and standard error σ
S1
S2
ˆ
X
±
X
,
σ
1
2
S
±
S
1
2
and X
are the values of the statistics S
1
2
and S
1
2
⋅σ
, p'+z
α
α
/2
p'
⋅σ
α
n-1,
/2
and µ
S1
S1
S2
. Also, the sum of the statistics
= (σ
S1+S2
2
2
σ
σ
=
S
1
+
S
2
n
n
1
2
and S
1
2
and σ
2
are the variances
S1
S2
from which the samples were
Page 18-25
⋅σ
).
/2
p'
,p'+t
p'
n-
,
S2
= µ
-
S1
2
+
S1
from
2  