FORMULA CARD FOR WEISS’S ELEMENTARY STATISTICS, FOURTH EDITION
Larry R. Griffey
• Linear correlation coefficient:
1
6(x − x)(y − y)
r H n−1
sx sy
NOTATION In the formulas below, unless stated otherwise, we
employ the following notation which may or may not appear with
subscripts:
σ
d
p̂
p
O
E
n H sample size
x H sample mean
s H sample stdev
Qj H j th quartile
N H population size
µ H population mean
H population stdev
H paired difference
H sample proportion
H population proportion
H observed frequency
H expected frequency
Sxy
rH p
Sxx Syy
CHAPTER 5 Probability and Random Variables
• Probability for equally likely outcomes:
f
,
P (E) H
N
where f denotes the number of ways event E can occur and
N denotes the total number of outcomes possible.
CHAPTER 3 Descriptive Measures
• Special addition rule:
6x
• Sample mean: x H
n
P (A or B or C or · · · ) H P (A) + P (B) + P (C) + · · ·
(A, B, C, . . . mutually exclusive)
• Range: Range H Max − Min
• Complementation rule: P (E) H 1 − P (not E)
• Sample standard deviation:
s
6(x − x)2
or
sH
n−1
• Quartile positions: (n + 1)/4,
s
sH
6x 2 − (6x)2 /n
n−1
(n + 1)/2,
3(n + 1)/4
• General addition rule: P (A or B) H P (A) + P (B) − P (A & B)
• Mean of a discrete random variable X: µ H 6xP (X H x)
• Standard deviation of a discrete random variable X:
p
p
σ H 6(x − µ)2 P (X H x) or σ H 6x 2 P (X H x) − µ2
• Interquartile range: IQR H Q3 − Q1
• Lower limit H Q1 − 1.5 · IQR,
• Factorial: k! H k(k − 1) · · · 2 · 1
n
n!
• Binomial coefficient:
H
x! (n − x)!
x
Upper limit H Q3 + 1.5 · IQR
• Population mean (mean of a variable): µ H
6x
N
• Binomial probability formula:
• Population standard deviation (standard deviation of a variable):
s
s
6(x − µ)2
6x 2
σ H
or
σ H
− µ2
N
N
• Standardized variable: z H
or
n x
P (X H x) H
p (1 − p)n−x ,
x
where n denotes the number of trials and p denotes the success
probability.
x−µ
σ
• Mean of a binomial random variable: µ H np
CHAPTER 4 Descriptive Methods in Regression and Correlation
• Standard deviation
p of a binomial random
variable: σ H np(1 − p)
• Sxx , Sxy , and Syy :
Sxx H 6(x − x)2 H 6x 2 − (6x)2 /n
CHAPTER 7 The Sampling Distribution of the Mean
Sxy H 6(x − x)(y − y) H 6xy − (6x)(6y)/n
• Mean of the variable x: µx H µ
Syy H 6(y − y)2 H 6y 2 − (6y)2 /n
√
• Standard deviation of the variable x: σx H σ/ n
• Regression equation: ŷ H b0 + b1 x, where
Sxy
1
b1 H
and
b0 H (6y − b1 6x) H y − b1 x
Sxx
n
• Standardized version of the variable x:
x−µ
zH
√
σ/ n
• Total sum of squares: SST H 6(y − y)2 H Syy
CHAPTER 8 Confidence Intervals for One Population Mean
2
• Regression sum of squares: SSR H 6(ŷ − y)2 H Sxy
/Sxx
• z-interval for µ (σ known, normal population or large sample):
σ
x ± zα/2 · √
n
2
• Error sum of squares: SSE H 6(y − ŷ)2 H Syy − Sxy
/Sxx
• Regression identity: SST H SSR + SSE
• Coefficient of determination: r 2 H
σ
• Margin of error for the estimate of µ: E H zα/2 · √
n
SSR
SST
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FORMULA CARD FOR WEISS’S ELEMENTARY STATISTICS, FOURTH EDITION
Larry R. Griffey
• Nonpooled t-test statistic for H0 : µ1 H µ2 (independent samples,
and normal populations or large samples):
x1 − x2
tH q
2
(s1 /n1 ) + (s22 /n2 )
• Sample size for estimating µ:
zα/2 · σ 2
,
nH
E
rounded up to the nearest whole number.
with df H 1.
• Studentized version of the variable x:
x−µ
tH
√
s/ n
• Nonpooled t-interval for µ1 − µ2 (independent samples, and
normal populations or large samples):
q
(x 1 − x 2 ) ± tα/2 · (s12 /n1 ) + (s22 /n2 )
• t-interval for µ (σ unknown, normal population or large sample):
s
x ± tα/2 · √
n
with df H 1.
• Paired t-test statistic for H0 : µ1 H µ2 (paired sample, and normal
differences or large sample):
with df H n − 1.
CHAPTER 9 Hypothesis Tests for One Population Mean
tH
• z-test statistic for H0 : µ H µ0 (σ known, normal population or
large sample):
zH
with df H n − 1.
• Paired t-interval for µ1 − µ2 (paired sample, and normal differences or large sample):
sd
d ± tα/2 · √
n
with df H n − 1.
x − µ0
√
σ/ n
• t-test statistic for H0 : µ H µ0 (σ unknown, normal population or
large sample):
tH
d
√
sd / n
x − µ0
√
s/ n
CHAPTER 11 Inferences for Population Proportions
• Sample proportion:
with df H n − 1.
x
,
n
where x denotes the number of successes.
p̂ H
CHAPTER 10 Inferences for Two Population Means
• Pooled sample standard deviation:
s
(n1 − 1)s12 + (n2 − 1)s22
sp H
n1 + n2 − 2
• One-sample z-interval for p:
p̂ ± zα/2 ·
p
p̂(1 − p̂)/n
(Assumption: both x and n − x are 5 or greater)
• Pooled t-test statistic for H0 : µ1 H µ2 (independent samples, normal populations or large samples, and equal population standard
deviations):
x1 − x2
tH √
sp (1/n1 ) + (1/n2 )
with df H n1 + n2 − 2.
• Margin of error for the estimate of p:
p
E H zα/2 · p̂(1 − p̂)/n
• Sample size for estimating p:
zα/2 2
zα/2 2
n H 0.25
or
n H p̂g (1 − p̂g )
E
E
rounded up to the nearest whole number (g H “educated guess”)
• Pooled t-interval for µ1 − µ2 (independent samples, normal
populations or large samples, and equal population standard
deviations):
p
(x 1 − x 2 ) ± tα/2 · sp (1/n1 ) + (1/n2 )
• One-sample z-test statistic for H0 : p H p0 :
p̂ − p0
zH √
p0 (1 − p0 )/n
(Assumption: both np0 and n(1 − p0 ) are 5 or greater)
with df H n1 + n2 − 2.
• Pooled sample proportion: p̂p H
• Degrees of freedom for nonpooled-t procedures:
i2
h
s12 /n1 + s22 /n2
1H 2
2 ,
s12 /n1
s22 /n2
+
n1 − 1
n2 − 1
rounded down to the nearest integer.
x1 + x2
n1 + n2
• Two-sample z-test statistic for H0 : p1 H p2 :
p̂1 − p̂2
zH p
√
p̂p (1 − p̂p ) (1/n1 ) + (1/n2 )
(Assumptions: independent samples; x1 , n1 − x1 , x2 , n2 − x2 are
all 5 or greater)
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FORMULA CARD FOR WEISS’S ELEMENTARY STATISTICS, FOURTH EDITION
Larry R. Griffey
• Two-sample z-interval for p1 − p2 :
p
(p̂1 − p̂2 ) ± zα/2 · p̂1 (1 − p̂1 )/n1 + p̂2 (1 − p̂2 )/n2
• One-way ANOVA identity: SST H SSTR + SSE
• Computing formulas for sums of squares in one-way ANOVA:
(Assumptions: independent samples; x1 , n1 − x1 , x2 , n2 − x2 are
all 5 or greater)
SST H 6x 2 − (6x)2 /n
SSTR H 6(Tj2 /nj ) − (6x)2 /n
• Margin of error for the estimate of p1 − p2 :
p
E H zα/2 · p̂1 (1 − p̂1 )/n1 + p̂2 (1 − p̂2 )/n2
SSE H SST − SSTR
• Mean squares in one-way ANOVA:
• Sample size for estimating p1 − p2 :
zα/2 2
n1 H n2 H 0.5
E
or
2
z
α/2
n1 H n2 H p̂1g (1 − p̂1g ) + p̂2g (1 − p̂2g )
E
rounded up to the nearest whole number (g H “educated guess”)
MSTR H
SSTR
,
k−1
MSE H
SSE
n−k
• Test statistic for one-way ANOVA (independent samples, normal
populations, and equal population standard deviations):
F H
MSTR
MSE
with df H (k − 1, n − k).
CHAPTER 12 Chi-Square Procedures
CHAPTER 14 Inferential Methods in Regression and Correlation
• Expected frequencies for a chi-square goodness-of-fit test:
E H np
• Population regression equation: y H β0 + β1 x
r
SSE
• Standard error of the estimate: se H
n−2
• Test statistic for a chi-square goodness-of-fit test:
χ 2 H 6(O − E)2 /E
with df H k − 1, where k is the number of possible values for the
variable under consideration.
• Test statistic for H0 : β1 H 0:
tH
• Expected frequencies for a chi-square independence test:
R·C
EH
n
where R H row total and C H column total.
with df H n − 2.
• Confidence interval for β1 :
se
b1 ± tα/2 · √
Sxx
• Test statistic for a chi-square independence test:
with df H n − 2.
χ 2 H 6(O − E)2 /E
with df H (r − 1)(c − 1), where r and c are the number of possible
values for the two variables under consideration.
• Confidence interval for the conditional mean of the response
variable corresponding to xp :
s
(xp − 6x/n)2
1
+
ŷp ± tα/2 · se
n
Sxx
CHAPTER 13 Analysis of Variance (ANOVA)
• Notation in one-way ANOVA:
with df H n − 2.
k H number of populations
n H total number of observations
x H mean of all n observations
nj H size of sample from Population j
x j H mean of sample from Population j
sj2 H variance of sample from Population j
Tj H sum of sample data from Population j
• Prediction interval for an observed value of the response variable
corresponding to xp :
s
(xp − 6x/n)2
1
+
ŷp ± tα/2 · se 1 +
n
Sxx
with df H n − 2.
• Test statistic for H0 : ρ H 0:
• Defining formulas for sums of squares in one-way ANOVA:
SST H 6(x − x)
b1
√
se / Sxx
tH s
2
SSTR H 6nj (x j − x)
2
with df H n − 2.
SSE H 6(nj − 1)sj2
-3-
r
1 − r2
n−2