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IMPORTANT FORMULAS
Chapter 3: Numerical Summaries of Data
Sample
mean:
x
x̄ =
n
Coefficient of variation:
σ
CV =
μ
Population
mean:
x
μ=
N
Range:
Range = largest value − smallest value
z-score:
x −μ
z=
σ
Interquartile range:
IQR = Q 3 − Q 1 = third quartile − first quartile
Population
variance:
(x − μ)2
σ2 =
N
Sample
variance:
(x − x̄)2
2
s =
n−1
Lower outlier boundary:
Q 1 − 1.5 IQR
Upper outlier boundary:
Q 3 + 1.5 IQR
Chapter 4: Probability
General Addition Rule:
P(A or B) = P(A) + P(B) − P(A and B)
General Method for Computing Conditional
Probability:
P(A and B)
P(B | A) =
P(A)
Multiplication Rule for Independent Events:
P(A and B) = P(A)P(B)
General Multiplication Rule:
P(A and B) = P(A)P(B | A) = P(B)P(A | B)
Addition Rule for Mutually Exclusive Events:
Permutation of r items chosen from n:
n!
n Pr =
(n − r )!
P(A or B) = P(A) + P(B)
Rule of Complements:
P(Ac ) = 1 − P(A)
Combination of r items chosen from n:
n!
n Cr =
r !(n − r )!
Chapter 5: Discrete Probability Distributions
Mean of a discrete random variable:
μ X = [x · P(x)]
Mean of a binomial random variable:
μ X = np
Standard
deviation of a discrete random variable:
σ X = σ X2
Standard
deviation of a binomial random variable:
σ X = np(1 − p)
Variance
variable:
of a discrete random
σ X2 = [(x − μ X )2 · P(x)] = [x 2 · P(x)] − μ2X
Variance of a binomial random variable:
σ X2 = np(1 − p)
Chapter 6: The Normal Distribution
z-score:
x −μ
z=
σ
z-score for a sample mean:
x̄ − μ
z=
σx̄
Convert z-score to raw score:
x = μ + zσ
Standard
deviation of the sample proportion:
p(1 − p)
σ p̂ =
n
Standard deviation of the sample mean:
σ
σx̄ = √
n
z-score for a sample proportion:
p̂ − p
z=
σ p̂
Chapter 7: Confidence Intervals
Confidence interval for a mean, standard deviation
known:
σ
σ
x̄ − z α/2 √ < μ < x̄ + z α/2 √
n
n
Confidence interval for a proportion:
Sample size to construct an interval for μ with margin
of error m:
z · σ 2
α/2
n=
m
Sample size to construct an interval for p with
margin of error m:
z 2
α/2
n = p̂(1 − p̂)
if a value for p̂ is available
m
Confidence interval for a mean, standard deviation
unknown:
s
s
x̄ − tα/2 √ < μ < x̄ + tα/2 √
n
n
p̂ − z α/2
n = 0.25
p̂(1 − p̂)
< p < p̂ + z α/2
n
z
α/2
m
2
p̂(1 − p̂)
n
if no value for p̂ is available
Chapter 8: Hypothesis Testing
Test statistic for a mean, standard deviation known:
x̄ − μ0
√
z=
σ/ n
Test statistic for a proportion:
p̂ − p0
z= p0 (1 − p0 )
n
Test statistic for a mean, standard deviation unknown:
x̄ − μ0
√
t=
s/ n
Chapter 9: Inferences on Two Samples
Test statistic for the difference between two means,
independent samples:
(x̄1 − x̄2 ) − (μ1 − μ2 )
t=
s12
s2
+ 2
n1
n2
Confidence interval for the difference between
two means, independent samples:
x̄1 − x̄2 − tα/2
s12
s2
+ 2 < μ1 − μ2
n1
n2
< x̄ 1 − x̄2 + tα/2
s12
s2
+ 2
n1
n2
Test statistic for the difference between
two proportions:
p̂1 − p̂2
z= 1
1
p̂(1 − p̂)
+
n1
n2
x1 + x2
where p̂ is the pooled proportion p̂ =
n1 + n2
Confidence interval for the difference between
two proportions:
p̂1 (1 − p̂1 )
p̂2 (1 − p̂2 )
p̂1 − p̂2 − z α/2
+
< p1 − p 2
n1
n2
< p̂1 − p̂2 + z α/2
p̂1 (1 − p̂1 )
p̂2 (1 − p̂2 )
+
n1
n2
Test statistic for the difference between two means,
matched pairs:
t=
d¯ − μ0
√
sd/ n
Confidence interval for the difference between two
means, matched pairs:
sd
sd
d¯ − tα/2 √ < μd < d¯ + tα/2 √
n
n
Chapter 10: Tests with Qualitative Data
Chi-square statistic:
(O − E)2
χ2 =
E
Expected frequency for goodness-of-fit:
E = np
Expected frequency for independence or homogeneity:
Row total · Column total
E=
Grand total
Chapter 11: Correlation and Regression
Correlation coefficient:
y − ȳ
1 x − x̄
r=
n−1
sx
sy
Equation of least-squares regression line:
ŷ = b0 + b1 x
Slope of least-squares regression line:
sy
b1 = r
sx
y-intercept of least-squares regression line:
b0 = ȳ − b1 x̄
Residual
standard deviation:
(y − ŷ)2
se =
n−2
Standard error for b1 :
se
sb = (x − x̄)2
Confidence interval for slope:
b1 − tα/2 · sb < β1 < b1 + tα/2 · sb
Test statistic for slope b1 :
b1
sb
t=
Confidence interval for the mean response:
ŷ ± tα/2 · se
1
(x ∗ − x̄)2
+
n
(x − x̄)2
Prediction interval
for an individual response:
ŷ ± tα/2 · se
1+
1
(x ∗ − x̄)2
+
n
(x − x̄)2