TESTBANKSEAGLES
, TESTBANKSEAGLES
This page intentionally left blank
, TESTBANKSEAGLES
Important Formulas
Chapter 3 Data Description Chapter 5 Discrete Probability Distributions
X Mean for a probability distribution: m [X P(X)]
Mean for individual data: X
n Variance and standard deviation for a probability
distribution:
f • Xm
Mean for grouped data: X s2 [X 2 P(X)] m2
n
Standard deviation for a sample: s [X 2 • PX ] m2
Expectation: E(X) [X P(X)]
X X 2 n X 2 X 2 n!
s or s Binomial probability: PX • pX • q nX
n1 nn 1 X !X!n
(Shortcut formula)
Mean for binomial distribution: m n p
Standard deviation for grouped data: Variance and standard deviation for the binomial
distribution: s2 n p q s n • p • q
n f • X m2 f • Xm 2 Multinomial probability:
s
nn 1 n!
PX • p X 1 • p2X 2 • p3X 3 • • • pkX k
range X1!X2!X3! . . . Xk! 1
Range rule of thumb: s
4 e X
Poisson probability: P(X; l) where
X!
Chapter 4 Probability and Counting Rules X 0, 1, 2, . . .
CX • bCnX
Addition rule 1 (mutually exclusive events): Hypergeometric probability: PX a
abCn
P(A or B) P(A) P(B)
Chapter 6 The Normal Distribution
Addition rule 2 (events not mutually exclusive):
P(A or B) P(A) P(B) P(A and B) X XX
Standard score z or z
s
Multiplication rule 1 (independent events):
Mean of sample means: mX m
P(A and B) P(A) P(B)
Standard error of the mean: sX
Multiplication rule 2 (dependent events): n
X
P(A and B) P(A) P(B A) Central limit theorem formula: z
n
P A and B
Conditional probability: PB A Chapter 7 Confidence Intervals and Sample
P A
Size
Complementary events: P(E ) 1 P(E) z confidence interval for means:
Fundamental counting rule: Total number of outcomes
of a sequence when each event has a different
X z 2 n
X z 2 n
number of possibilities: k 1 k 2 k 3 k n t confidence interval for means:
Permutation rule: Number of permutations of n objects
taking r at a time is n Pr
n!
X t 2 s n
X t 2 s n
n r !
z 2 • 2
Combination rule: Number of combinations of r objects Sample size for means: n
E where E is the
n! maximum error of estimate
selected from n objects is n Cr
n r !r!
Confidence interval for a proportion:
p̂ q̂ p̂ q̂
p̂ z 2 p p̂ z 2
n n
, TESTBANKSEAGLES
2
z 2
Sample size for a proportion: n p̂ q̂ E Formula for the confidence interval for difference of two
means (small independent samples, variance
X unequal):
where p̂ and q̂ 1 p̂
n
s21 s22
Confidence interval for variance: X1 X2 t 2 1 2
n1 n2
n 1 s2 n 1 s2
2
s21 s22
2right 2left X1 X2 t 2
n1 n2
Confidence interval for standard deviation:
(d.f. smaller of n1 1 and n2 1)
n 1 s2 n 1 s2
t test for comparing two means for dependent samples:
2right 2left
D D D
t where D and
sD n n
Chapter 8 Hypothesis Testing
X nD 2 D 2
z test: z for any value n. If n 30, sD d.f. n 1
n nn 1
population must be normally distributed.
Formula for confidence interval for the mean of the
X difference for dependent samples:
t test: t (d.f. n 1)
s n SD SD
p̂ p D t 2 D D t 2
z test for proportions: z n n
pq n
(d.f. n 1)
n 1 s 2
Chi-square test for a single variance: 2 2 z test for comparing two proportions:
(d.f. n 1) p̂1 p̂2 p1 p2
z
n1 n1
__
pq
Chapter 9 Testing the Difference Between 1 2
Two Means, Two Proportions, _ X1 X2 X1
and Two Variances where p p̂1
n1 n2 n1
z test for comparing two means (independent samples): _ _ X2
q1p p̂2
X1
X2 n2
1 2
z
2
1
2
2 Formula for the confidence interval for the difference of
two proportions:
n1 n2
Formula for the confidence interval for difference of two p̂1 q̂1 p̂2 q̂2
p̂1 p̂2 z 2 p1 p2
means (large samples): n1 n2
2
1
2
2 p̂1 q̂1 p̂2 q̂2
X1 X2 z 2 1 2
p̂1 p̂2 z 2
n1 n2 n1 n2
s21
2 2
X1 X2 z 2 1
2 F test for comparing two variances: F where s 21 is the
n1 n2 s22
larger variance and d.f.N. n1 1, d.f.D. n2 1
t test for comparing two means (independent samples,
variances not equal):
X1 X2 1 2
t
s21 s22
n1 n2
(d.f. the smaller of n 1 1 or n2 1)
, TESTBANKSEAGLES
This page intentionally left blank
, TESTBANKSEAGLES
Important Formulas
Chapter 3 Data Description Chapter 5 Discrete Probability Distributions
X Mean for a probability distribution: m [X P(X)]
Mean for individual data: X
n Variance and standard deviation for a probability
distribution:
f • Xm
Mean for grouped data: X s2 [X 2 P(X)] m2
n
Standard deviation for a sample: s [X 2 • PX ] m2
Expectation: E(X) [X P(X)]
X X 2 n X 2 X 2 n!
s or s Binomial probability: PX • pX • q nX
n1 nn 1 X !X!n
(Shortcut formula)
Mean for binomial distribution: m n p
Standard deviation for grouped data: Variance and standard deviation for the binomial
distribution: s2 n p q s n • p • q
n f • X m2 f • Xm 2 Multinomial probability:
s
nn 1 n!
PX • p X 1 • p2X 2 • p3X 3 • • • pkX k
range X1!X2!X3! . . . Xk! 1
Range rule of thumb: s
4 e X
Poisson probability: P(X; l) where
X!
Chapter 4 Probability and Counting Rules X 0, 1, 2, . . .
CX • bCnX
Addition rule 1 (mutually exclusive events): Hypergeometric probability: PX a
abCn
P(A or B) P(A) P(B)
Chapter 6 The Normal Distribution
Addition rule 2 (events not mutually exclusive):
P(A or B) P(A) P(B) P(A and B) X XX
Standard score z or z
s
Multiplication rule 1 (independent events):
Mean of sample means: mX m
P(A and B) P(A) P(B)
Standard error of the mean: sX
Multiplication rule 2 (dependent events): n
X
P(A and B) P(A) P(B A) Central limit theorem formula: z
n
P A and B
Conditional probability: PB A Chapter 7 Confidence Intervals and Sample
P A
Size
Complementary events: P(E ) 1 P(E) z confidence interval for means:
Fundamental counting rule: Total number of outcomes
of a sequence when each event has a different
X z 2 n
X z 2 n
number of possibilities: k 1 k 2 k 3 k n t confidence interval for means:
Permutation rule: Number of permutations of n objects
taking r at a time is n Pr
n!
X t 2 s n
X t 2 s n
n r !
z 2 • 2
Combination rule: Number of combinations of r objects Sample size for means: n
E where E is the
n! maximum error of estimate
selected from n objects is n Cr
n r !r!
Confidence interval for a proportion:
p̂ q̂ p̂ q̂
p̂ z 2 p p̂ z 2
n n
, TESTBANKSEAGLES
2
z 2
Sample size for a proportion: n p̂ q̂ E Formula for the confidence interval for difference of two
means (small independent samples, variance
X unequal):
where p̂ and q̂ 1 p̂
n
s21 s22
Confidence interval for variance: X1 X2 t 2 1 2
n1 n2
n 1 s2 n 1 s2
2
s21 s22
2right 2left X1 X2 t 2
n1 n2
Confidence interval for standard deviation:
(d.f. smaller of n1 1 and n2 1)
n 1 s2 n 1 s2
t test for comparing two means for dependent samples:
2right 2left
D D D
t where D and
sD n n
Chapter 8 Hypothesis Testing
X nD 2 D 2
z test: z for any value n. If n 30, sD d.f. n 1
n nn 1
population must be normally distributed.
Formula for confidence interval for the mean of the
X difference for dependent samples:
t test: t (d.f. n 1)
s n SD SD
p̂ p D t 2 D D t 2
z test for proportions: z n n
pq n
(d.f. n 1)
n 1 s 2
Chi-square test for a single variance: 2 2 z test for comparing two proportions:
(d.f. n 1) p̂1 p̂2 p1 p2
z
n1 n1
__
pq
Chapter 9 Testing the Difference Between 1 2
Two Means, Two Proportions, _ X1 X2 X1
and Two Variances where p p̂1
n1 n2 n1
z test for comparing two means (independent samples): _ _ X2
q1p p̂2
X1
X2 n2
1 2
z
2
1
2
2 Formula for the confidence interval for the difference of
two proportions:
n1 n2
Formula for the confidence interval for difference of two p̂1 q̂1 p̂2 q̂2
p̂1 p̂2 z 2 p1 p2
means (large samples): n1 n2
2
1
2
2 p̂1 q̂1 p̂2 q̂2
X1 X2 z 2 1 2
p̂1 p̂2 z 2
n1 n2 n1 n2
s21
2 2
X1 X2 z 2 1
2 F test for comparing two variances: F where s 21 is the
n1 n2 s22
larger variance and d.f.N. n1 1, d.f.D. n2 1
t test for comparing two means (independent samples,
variances not equal):
X1 X2 1 2
t
s21 s22
n1 n2
(d.f. the smaller of n 1 1 or n2 1)
