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Chapter 3 Data Description
Mean for individual data:
Mean for grouped data:
Standard deviation for a sample:
or
(Shortcut formula)
Standard deviation for grouped data:
Range rule of thumb:
Chapter 4 Probability and Counting Rules
Addition rule 1 (mutually exclusive events):
P(A or B) P(A) P(B)
Addition rule 2 (events not mutually exclusive):
P(A or B) P(A) P(B) P(A and B)
Multiplication rule 1 (independent events):
P(A and B) P(A) P(B)
Multiplication rule 2 (dependent events):
P(A and B) P(A) P(B A)
Conditional probability:
Complementary events: P( ) 1 P(E)
Fundamental counting rule: Total number of outcomes of a sequence when each event has a different number of possibilities: k 1 k 2 k 3 k (^) n
Permutation rule: Number of permutations of n objects
taking r at a time is
Combination rule: Number of combinations of r objects
selected from n objects is (^) n Cr
n! n r!r!
n Pr ^
n! n r!
E
PB (^) A^
PA and B PA
s
range 4
s
n f • X (^2) m f • Xm ^2 nn 1
s
nX 2 X^2 nn 1
s
X X^2
n 1
X^ ^
f • Xm n
X
X
n
Chapter 5 Discrete Probability Distributions
Mean for a probability distribution: m [X P(X)] Variance and standard deviation for a probability distribution: s^2 [X^2 P(X)] m^2
Expectation: E(X) [X P(X)]
Binomial probability:
Mean for binomial distribution: m n p Variance and standard deviation for the binomial distribution: s^2 n p q s Multinomial probability:
Poisson probability: P(X; l) where X 0, 1, 2,... Hypergeometric probability:
Chapter 6 The Normal Distribution
Standard score
Mean of sample means: mX m
Standard error of the mean: sX
Central limit theorem formula:
Chapter 7 Confidence Intervals and Sample
Size
z confidence interval for means:
t confidence interval for means:
Sample size for means: where E is the
maximum error of estimate Confidence interval for a proportion:
pˆ z 2 ^
pˆ qˆ n
p pˆ z 2 ^
pˆ qˆ n
n
z 2 •
E
2
X
t 2
s
n^
X
t 2
s
n
X^ ^ z 2
n^
X^ ^ z 2
n
z
X^ ^
n
n
z
X
or z
X X^
s
PX^ a
CX • (^) bCnX abC^ n
e^ X X!
PX
n! X 1 !X 2 !X 3!...^ Xk!
- p 1 X 1 • p X 2 2 • p X 3 3 • • • p (^) kX^ k
n^ •^ p^ •^ q
PX^
n! n X!X! •^ p
X (^) • q nX
s [X 2 • PX] m^2
Important Formulas
Sample size for a proportion:
where
Confidence interval for variance:
Confidence interval for standard deviation:
Chapter 8 Hypothesis Testing
z test: for any value n. If n 30,
population must be normally distributed.
t test: (d.f. n 1)
z test for proportions:
Chi-square test for a single variance:
(d.f. n 1)
Chapter 9 Testing the Difference Between
Two Means, Two Proportions,
and Two Variances
z test for comparing two means (independent samples):
Formula for the confidence interval for difference of two means (large samples):
t test for comparing two means (independent samples, variances not equal):
(d.f. the smaller of n 1 1 or n 2 1)
t
X^ 1 X^ 2 1 2
s^21 n 1
s
(^22) n 2
X^ 1 X^ 2 z 2
(^21) n 1
2 n 2
X^ 1 X^ 2 (^) z 2
(^21) n 1
2 n 2 1
z
X^ 1 X^ 2 1 2
12 n 1 ^
(^22) n 2
n 1 s^2 2
z
pˆ p pq n
t
X^ ^
s (^) n
z
X
n
n 1 s^2 (^2) right
n 1 s^2 (^2) left
n 1 s^2 (^2) right
2 n^ ^1 s
2 (^2) left
pˆ X n
and qˆ 1 pˆ
n pˆ qˆ
z 2 E
(^2) Formula for the confidence interval for difference of two means (small independent samples, variance unequal):
(d.f. smaller of n 1 1 and n 2 1) t test for comparing two means for dependent samples:
Formula for confidence interval for the mean of the difference for dependent samples:
(d.f. n 1) z test for comparing two proportions:
where
Formula for the confidence interval for the difference of two proportions:
F test for comparing two variances: where is the
larger variance and d.f.N. n 1 1, d.f.D. n 2 1
F s^21
s^21 s^22
pˆ 1 pˆ 2 z 2
pˆ 1 qˆ 1 n 1
pˆ 2 qˆ 2 n 2
(^) pˆ 1 pˆ 2 (^) z 2
pˆ 1 qˆ 1 n 1 ^
pˆ 2 qˆ 2 n 2 p^1 ^ p^2
q
_ 1 p
_ p ˆ 2
X 2
n 2
p
_
X 1 X 2
n 1 n 2
p ˆ 1
X 1
n 1
z
(^) pˆ 1 pˆ 2 (^) (^) p 1 p 2
p
_ q
_
n 1
n 2
D^ ^ t 2
S D
n^ D^
D^ ^ t 2
S D
n
sD
nD 2 D^2 nn 1 ^
d.f. n 1
t D
(^) D s (^) D n
where D^ ^ D n
and
X^ 1 X^ 2 (^) t 2
s^21 n 1
s^22 n 2
X^ 1 X^ 2 (^) t 2
s^21 n 1
s^22 n 2 1
Chapter 13 Nonparametric Statistics
z test value in the sign test:
where n sample size (greater than or equal to 26) X smaller number of or signs
Wilcoxon rank sum test:
where
R sum of the ranks for the smaller sample size (n 1 ) n 1 smaller of the sample sizes n 2 larger of the sample sizes n 1 10 and n 2 10
Wilcoxon signed-rank test:
where
n number of pairs where the difference is not 0 ws smaller sum in absolute value of the signed ranks
z
ws
nn 1 4
A
nn 1 2 n 1 24
R ^
n 1 n 2 n 1 n 2 1 12
R ^
n 1 n 1 n 2 1 2
z
R mR sR
z
X 0.5 (^) n 2 n^2
Kruskal-Wallis test:
where R 1 sum of the ranks of sample 1 n 1 size of sample 1 R 2 sum of the ranks of sample 2 n 2 size of sample 2 Rk sum of the ranks of sample k nk size of sample k N n 1 n 2 n (^) k k number of samples Spearman rank correlation coefficient:
where d difference in the ranks n number of data pairs
rS 1
6 d 2 nn^2 1
H
NN 1 ^
R^21
n 1
R^22
n 2
R^2 k nk ^
3 N 1
ISBN-13: 978–0–07–743861–6ISBN-10: 0–07–743861–
Procedure Table
Solving Hypothesis-Testing Problems (Traditional Method)
Step 1 State the hypotheses and identify the claim.
Step 2 Find the critical value(s) from the appropriate table in Appendix C.
Step 3 Compute the test value.
Step 4 Make the decision to reject or not reject the null hypothesis.
Step 5 Summarize the results.
Procedure Table
Solving Hypothesis-Testing Problems (P-value Method)
Step 1 State the hypotheses and identify the claim.
Step 2 Compute the test value.
Step 3 Find the P-value.
Step 4 Make the decision.
Step 5 Summarize the results.
