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A Statistics Summary-sheet
Sampling Conditions
Confidence Interval
Test Statistic
(^2) σis known
⇒^
X^ ∼^
N (μ
(^2) , σ/n)
±^
n Z X
σ α^2 /
n X Z^
μ−^0 / σ
Yes
(^2) σis unknown
⇒^
X^ ∼
N (μ
(^2) , σ/n)
±^
s n Z X^
(^2) / α
ns X Z^
μ−^0 /
Is n is large, say over 30?
^
−
±^
pp n
Zp
) (^1) ( (^2) / α
pp pp n Z^
) (^1) (
0 0 −^0 − =
No^
X^ ∼^
N (μ
(^2) , σ) and
(^2) σis known
⇒^
X^ ∼
N (μ
(^2) , σ/n)
±^
n Z X
σ α^2 /
n X Z^
μ−^0 / σ
X^ ∼^
N (μ
(^2) , σ) and
(^2) σis unknown
⇒^
X^ ∼
tn-^
(μ,^ σ
2 /n)^
±^ −
s^ n t X^
n^
(^2) /, 1 α
ns X tn
(^0) / 1
μ− =−
If n is not large, say over 30 and X is not
∼∼∼∼^ N (
μμμμ ,^ σσσσ
2 ), cannot proceed with parametric statistics.
Formulas, Distributions, and Concepts
Counting and Probabilities
)!x !n n( Pxn
− =^
Permutations )!x !n n(!x
Cxn
− =^
Combinations )B( P
)B A(P )B| A(P
∩ =^
Conditional Probability^ )B(P) B|A (P )B A(P
= ∩^
Probability of an Intersection
Discrete Probability Distributions
xn x
x^
)p (^1) (p )!x !n n(!x )x( P^
− − − =^
Binomial Probability
! )( e x xP
x x
μ−^ μ =^
Poisson Probability
Continuous Probability Distributions Random Variable
∼^ Distribution (mean, variance)
Standard Normal
Z^ ∼
N(0,1)
s n
t X^
n^
) (^2) / , 1 (
α− ±^
If population is normal, population variance is unknown. p n p z p^
) (^2) / (
α^
If n^
≥30.^222 (^211) ) (^2) / ( ) (^
n n z Y X
α^
−^
If independent samples and either population variance known, or n
≥30 in which case, substitute sample
variance for population variance.
^
−^
−
2 1 2 ) (^2) / , 2
(
(^
n n s
t Y X^
Y nnX
α^
where
2 1
(^22) 2 (^21) 1 2
=^
n n
s n s n s^
If independent samples, population variances unknown, but statistically equal.
Estimating Sample Size
22 /2 2 E
z n
=^
For estimation and CI for the population mean, normal population,
(^2) σknown, or estimated by a pilot run. E = absolute error.
Hypothesis Testing 1. Set up the
appropriate
null which must be in equality form, always and alternative hypotheses.
- Define the rejection area. Take care as to whether the test is one-tailed or two-tailed. Look to the alternative hypothesis todetermine this.3. Calculate the test statistic.4. State Decision.5. Interpret your conclusion. Hypothesis Testing
(test statistics and their distributions under the null)
-^ n X^
0
μ^
∼ z^^ α
When population variance known, or if n
≥30, substitute
s^ for
σ.
-^ n X^
0 μ s ∼
tn-1,
When If population is normal, population variance unknown.α
or if the treatment sample sizes are all equal,
X k X
k j
j
∑^ =^1 =
(^1) − SSTR = k MSTR
(^
∑^ =
− k = j
j j^
X Xn
SSTR
1
2
k SSEn MSE
− T =^
∑^ =
− k = j
j j^
s n
SSE
1
(^2) ) 1 (
MSTR^ MSE F^ =
∼^
knT F^ k
− −^ ,^1
Terms and Concepts Central Limit Theorem: If the sample size
n^ is large, say n
≥30 no matter what the population distribution is, the sampling
distribution of the sample mean tends towards the normal as
n^ gets large.
