Download Chapter 2 Organizing and Summarizing Data

Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e ©2014 Pearson Education, Inc.­
Chapter 2 Organizing and Summarizing Data
• Relativefrequency =
frequency
• C
lass midpoint: The sum of consecutive lower class limits
divided by 2.
sumofallfrequencies
Chapter 3 Numerically Summarizing Data
• Population Mean: m =
gxi
n
• Sample Mean: x =
gxi
N
• Range = LargestDataValue - SmallestDataValue
( g xi)2
N
N
• Population Standard Deviation: g (xi - m)
=
B
R
N
s =
g x2i -
• Sample Standard Deviation s =
g(xi - x)2
B
n-1
=
gx2i
R
-
s=
( gxi)2
n-1
g(xi - m) fi
2
gfi
B
=
gx2ifi -
( gxifi)2
gfi
R
g(xi - m) f i
=
B ( gf i) - 1
R
x - m
• Population z-score: z =
s
x - x
• Sample z-score: z =
s
2
s =
• E
mpirical Rule: If the shape of the distribution is bellshaped, then
gfi
• Sample Standard Deviation from Grouped Data: n
• Sample Variance: s 2
gwi xi
gwi
• Weighted Mean: xw =
• Population Standard Deviation from Grouped Data: • Population Variance: s2
gxifi
gfi
• Sample Mean from Grouped Data: x =
2
gxifi
gfi
• Population Mean from Grouped Data: m =
• A
pproximately 68% of the data lie within 1 standard
deviation of the mean
• Approximately 95% of the data lie within 2 standard
deviations of the mean
• Approximately 99.7% of the data lie within 3 standard
deviations of the mean
gx 2if i -
( gxif i)2
gf i
gf i - 1
• Interquartile Range: IQR = Q 3 - Q 1
Lowerfence = Q1 - 1.5(IQR)
• Lower and Upper Fences:
Upperfence = Q3 + 1.5(IQR)
• Five-Number Summary
Minimum,Q 1 ,M,Q 3 ,Maximum
Chapter 4 Describing the Relation between Two Variables
• Correlation Coefficient: r =
aa
xi - x y i - y
ba
b
sx
sy
n-1
• The equation of the least-squares regression line is
yn = b1x + b0, where yn is the predicted value, b1 = r #
is the slope, and b0 = y - b1x is the intercept.
• Residual = observed y - predictedy = y - yn
• R 2 = r 2 for the least-squares regression model
yn = b1x + b0
sy
sx
• The coefficient of determination, R 2, measures the
proportion of total variation in the response variable that is explained by the least-squares regression line.
Chapter 5 Probability
• Empirical Probability
P(E) • Addition Rule for Disjoint Events
P(EorF ) = P(E) + P(F )
frequencyofE
numberoftrialsofexperiment
• Addition Rule for n Disjoint Events
P(EorForGor g) = P(E) + P(F ) + P(G) + g
• Classical Probability
P(E) =
numberofwaysthatEcanoccur
numberofpossibleoutcomes
=
N(E)
N(S)
• General Addition Rule
P(EorF ) = P(E) + P(F ) - P(EandF )
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Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e ©2014 Pearson Education, Inc.­
• Complement Rule
• Factorial
c
P(E ) = 1 - P(E)
• Multiplication Rule for Independent Events
• Permutation of n objects taken r at a time: nPr =
P(EandF ) = P(E) # P(F )
• Multiplication Rule for n Independent Events
P(EandFandG g ) = P(E) # P(F) # P(G) # g
• Permutations with Repetition:
• Conditional Probability Rule
P(F E) =
P(E)
=
n!
(n - r)!
• Combination of n objects taken r at a time:
n!
nC r =
r!(n - r)!
P(EandF )
n! = n # (n - 1) # (n - 2) # g # 3 # 2 # 1
n!
n1! # n2! # g # nk!
N(EandF )
N(E)
• General Multiplication Rule
P(EandF) = P(E) # P(F E)
Chapter 6 Discrete Probability Distributions
• Mean (Expected Value) of a Discrete Random Variable
• Binomial Probability Distribution Function
mX = gx # P(x)
P(x) = nCxpx(1 - p)n - x
• Standard Deviation of a Discrete Random Variable
• Mean and Standard Deviation of a Binomial Random Variable
sX = 3g(x - m)2 # P(x) = 3gx 2P(x) - m2X
mX = np sX = 2np(1 - p)
Chapter 7 The Normal Distribution
• Finding the Score: x = m + zs
• Standardizing a Normal Random Variable
x -m
z =
s
Chapter 8 Sampling Distributions
• M
ean and Standard Deviation of the Sampling Distribution
of x
s
mx = mandsx =
2n
x
• Sample Proportion: pn =
n
• M
ean and Standard Deviation of the Sampling Distribution of pn
mpn = pandspn =
p(1 - p)
B
n
Chapter 9 Estimating the Value of a Parameter
Confidence Intervals
Sample Size
• A (1 - a) # 100% confidence interval about m is x { ta/2 #
• T
o estimate the population proportion with a margin of error
E at a (1 - a) # 100% level of confidence:
za/2 2
b rounded up to the next integer,
n = pn (1 - pn ) a
E
where pn is a prior estimate of the population proportion,
za/2 2
or n = 0.25 a
b rounded up to the next integer when no
E
prior estimate of p is available.
• A (1 - a) # 100% confidence interval about p is
pn (1 - pn )
pn { za/2 #
.
B
n
Note: ta/2 is computed using n - 1 degrees of freedom.
s
.
1n
• To estimate the population mean with a margin of error E
za/2 # s 2
b
at a (1 - a) # 100% level of confidence: n = a
E
rounded up to the next integer.
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Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e ©2014 Pearson Education, Inc.­­­
Chapter 10 Hypothesis Tests Regarding a Parameter
Test Statistics
pn - p0
• z0 =
• t0 =
p0(1 - p0)
C
n
x - m0
s 1n
Chapter 11 Inferences on Two Samples
• Test Statistic Comparing Two Population Proportions
(Independent Samples)
z0 =
pn 1 - pn 2 - (p1 - p2)
where pn =
x1 + x2
.
n1 + n2
1
1
+
B n1 n2
• Confidence Interval for the Difference of Two Proportions
(Independent Samples)
2pn (1 - pn )
(pn 1 - pn 2) { za/2
pn 1(1 - pn 1)
C
n1
+
n2
(x1 - x2) - (m1 - m2)
s 21
s 22
+
C n1 n2
• C
onfidence Interval for the Difference of Two Means (Independent Samples)
s21
s22
+
C n1 n2
Note: ta/2 is found using the smaller of n1 - 1 or n2 - 1
degrees of freedom.
(x1 - x2) { ta/2
0 f12 - f21 0 - 1
2f 12 + f 21
• Test Statistic for Matched-Pairs Data
t0 =
t0 =
pn 2(1 - pn 2)
• Test Statistic Comparing Two Proportions (Dependent Samples)
z0 =
• Confidence Interval for Matched-Pairs Data
sd
d { ta/2 #
1n
Note: ta/2 is found using n - 1 degrees of freedom.
• Test Statistic Comparing Two Means (Independent Sampling)
d - md
sd 1n
where d is the mean and sd is the standard deviation of the
differenced data.
Chapter 12 Additional Inferential Procedures
Chi-Square Procedures
• Expected Counts (when testing for goodness of fit)
E i = mi = npifori = 1,2, p ,k
• Expected Frequencies (when testing for independence or
homogeneity of proportions)
Expectedfrequency =
(rowtotal)(columntotal)
tabletotal
• Chi-Square Test Statistic
x20 = a
(observed - expected)2
expected
=a
(Oi - E i)2
Ei
i = 1,2, p ,k
All E i Ú 1 and no more than 20% less than 5.
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Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e ©2014 Pearson Education, Inc.­­­
Inference on the Least-Squares Regression Model
• Standard Error of the Estimate
se =
g(yi - yn i)
C
n-2
2
=
• Standard error of b1
sb1 =
se
• Confidence Interval about the Mean Response of y,yn
g residuals
n- 2
2
C
yn { ta/2 # se
2g(xi - x)2
• Test statistic for the Slope of the Least-Squares Regression
Line
b1 - b1
b1 - b1
t0 =
=
se
sb1
n 2g(xi - x)2
• Confidence Interval for the Slope of the Regression Line
se
b1 { ta/2 #
2g(xi - x)2
(x* - x)2
1
+
C n g(xi - x)2
where x* is the given value of the explanatory variable and
ta/2 is the critical value with n - 2 degrees of freedom.
• Prediction Interval about an Individual Response, yn
yn { ta/2 # se
C
1 +
(x* - x)2
1
+
n
g(xi - x)2
where x* is the given value of the explanatory variable and
ta/2 is the critical value with n - 2 degrees of freedom.
where ta/2 is computed with n - 2 degrees of freedom.
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