STA 5166 Formula Sheet for Exam II
Tolerance interval which contains at least1
of all
future observations from a given population with some
preassigned probability 1
Measures for accuracy and precision:
Bias
̅
Var
, ̅
Find in Table A.12 for selected values of , 1
and 1
.
MSE
,
The pooled estimator of variance:
MSE
∑
∑
Probabilities of Type I and Type II errors:
Reject
|
Fail to reject
1
Power:
)
|
Reject
|
|
T
~
/
Inference for single sample means
~
/√
0,1
/
Sample size to ensure a 100 1
% z-interval width
W 2E where is the margin of error:
/
HT
Twosided
/
Power Function
Sample size
√
/
|
Upper
onesided
Lower
onesided
1
SEM ̅
SEM
and
1
/
When
and
, the minimum
size sample required to detect at least 1   2  
|
n
2
for two-sided test: n
√
/
difference with at least 1   power for one-sided test:
/
√
/
/
30
for large
2
~
The Welch-Satterthwaite method using separate variance
estimates:
)
Operating Characteristic Function:
Test fails to reject
1/
1/
)
/
2
Matched Pair Data:
∑
̅
/√
~
for small
Inference for single sample variance:
1
~
Prediction Interval for future observation:
T
~
1, ⋯ ,
,
√
t
/√
~
,
∑
for small sample size
Inference for the ratio of two variances:
/
~
/


Sample proportion pˆ  N  p,
,
p(1  p) 
 for large
n

1
STA 5166 Formula Sheet for Exam II
An approximate (1   ) -level CI for p is
̂
̂
/
∗
∗
Can test
∗
is a prior guess of p ; when
0: p
vs.
1: p
2
1:
1
2
0:
1
2
1
2
vs.
1:
1
/
0
√
1
:
:
2
1
0
1
1
1
:
0
:
for
2
1
0
0
1
0
1
1
0
|
,
|
P-value for small
:
1
|
vs.
1
2
|
vs.
1
2
:
0
vs.
1
2
0
1
/2
1
0
̂
-level two-sided CI for
̂
1
1
0
0
for
1
P-value for large
(z-test with
continuity correction)
1
1
√
1
√
∑
,
2
,
2
Chi-square test for two-way count data:
2
0
}
∑
for
/2
|
2min{
(Exact Test )
1
Sample Size Determination
0
|
,
HT based on
/
with small samples
McNemar’s test for matched pair Bernoulli samples
1
0
1
0
1
vs.
1
0 when
2
0:
1
√
0
1
1:
1
0
2
1
0
:
P-value
1
vs.
√
0
√
Reject
0:
Power Function
0
0: p
vs.
:
p
1
using
HT
/
for HT on p
/
:
Fisher’s exact test for
is not available, a
conservative upper bound is given by
HT
0: p
vs.
:
p
1
using
where ̂
∗
1
When using
:
vs.
/
Sample size to ensure a 100 1
% z-interval width
W 2E where is the margin of error:
/
:
Test
|
|
:
1
∑
∑
̂
̂
~
Pearson’s chi-square test:
∑
parameters in
~
,
= # of indep.
- # of indep. parameters estimated
/
2