F-1
Chapter 1
• The mean x. If the n observations are x1, x2, Á , xn, their
mean is
x 5
x1 1 x2 1 Á 1 xn
n
• The median M. Arrange all observations in order of size,
from smallest to largest. If the number of observations
n is odd, the median M is the center observation in the
ordered list. Find the location of the median by counting
(n 1 1)y2 observations up from the bottom of the list. If
the number of observations n is even, the median M is the
mean of the two center observations in the ordered list.
The location of the median is again (n 1 1)y2 from the
bottom of the list.
• The quartiles Q1 and Q3. Arrange the observations in
increasing order and locate the median M in the ordered
list of observations. Q1 is the median of the observations
whose position in the ordered list is to the left of the
location of the overall median. Q3 is the median of the
observations whose position in the ordered list is to the
right of the location of the overall median.
• The five-number summary. The smallest observation,
the first quartile, the median, the third quartile, and the
largest observation, written in order from smallest to
largest. In symbols, the five-number summary is
Minimum Q1 M Q3 Maximum
• A boxplot. A graph of the five-number summary. A central
box spans the quartiles Q1 and Q3. A line in the box marks
the median M. Lines extend from the box out to the
smallest and largest observations.
• The interquartile range(IQR.) The distance between the
first and third quartiles,
IQR 5 Q3 2 Q1
• The 1.5 3 IQR rule for outliers. Call an observation a
suspected outlier if it falls more than 1.5 3 IQR above the
third quartile or below the first quartile.
• The variance s2. For n observations x1, x2, Á , xn,
s2 5
(x1 2 x)
2 1 (x2 2 x)
2 1 Á 1 (xn 2 x)2
n 2 1
• The standard deviation s. is the square root of the
variance s2.
• Effect of a linear transformation. Multiplying each
observation by a positive number b multiplies both
measures of center (mean and median) and measures of
spread (interquartile range and standard deviation) by b.
Adding the same number a (either positive or negative)
to each observation adds a to measures of center and
to quartiles and other percentiles but does not change
measures of spread.
• Density curve. Is always on or above the horizontal axis
and has area exactly 1 underneath it.
• The median of a density curve. The equal-areas point,
the point that divides the area under the curve in half.
• The mean of a density curve. the balance point at which
the curve would balance if made of solid material.
• The 68–95–99.7 rule. In the Normal distribution
with mean and standard deviation , Approximately
68% of the observations fall within of the mean ,
approximately 95% of the observations fall within 2 of ,
and approximately 99.7% of the observations fall within
3 of .
• Standardizing and z-scores. If x is an observation
from a distribution that has mean and standard
deviation ,
z 5
x 2

• The standard Normal distribution. The Normal
distribution N(0, 1) with mean 0 and standard deviation 1.
If a variable X has any Normal distribution N(, ) with
mean and standard deviation , then the standardized
variable
Z 5
X 2

has the standard Normal distribution.
• Use of Normal quantile plots. If the points on a Normal
quantile plot lie close to a straight line, the plot indicates
that the data are Normal. Systematic deviations from a
straight line indicate a non-Normal distribution. Outliers
appear as points that are far away from the overall pattern
of the plot.
Chapter 2
• Response variable, explanatory variable. A response
variable measures an outcome of a study. An explanatory
variable explains or causes changes in the response variables.
Formulas and Key Ideas
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F-2 Formulas and Key Ideas
• Scatterplot. A scatterplot shows the relationship
between two quantitative variables measured on the
same individuals. The values of one variable appear on
the horizontal axis, and the values of the other variable
appear on the vertical axis. Each individual in the data
appears as the point in the plot fixed by the values of both
variables for that individual.
• Positive association, negative association. Two variables
are positively associated when above-average values of
one tend to accompany above-average values of the other
and below-average values also tend to occur together. Two
variables are negatively associated when above-average
values of one tend to accompany below-average values of
the other, and vice versa.
• Correlation. The correlation measures the direction
and strength of the linear relationship between two
quantitative variables. Correlation is usually written as
r. Suppose that we have data on variables x and y for n
individuals. The means and standard deviations of the two
variables are x and sx for the x-values, and y and sy for the
y-values. The correlation r between x and y is
r 5
1
n 2 1o1
xi 2 x
sx 21
yi 2 y
sy 2
• Straight lines. Suppose that y is a response variable
(plotted on the vertical axis) and x is an explanatory
variable (plotted on the horizontal axis). A straight line
relating y to x has an equation of the form
y 5 b0 1 b1x
In this equation, b1 is the slope, the amount by which y
changes when x increases by one unit. The number b0 is
the intercept, the value of y when x 5 0.
• Equation of the least-squares regression line. We
have data on an explanatory variable x and a response
variable y for n individuals. The means and standard
deviations of the sample data are x and sx for x and y
and sy for y, and the correlation between x and y is r. The
equation of the least-squares regression line of y on x is
y⁄ 5 b0 1 b1x
with slope b1 5 rsy ysx and intercept b0 5 y 2 b1x.
• r2 in regression. The square of the correlation, r2, is the
fraction of the variation in the values of y that is explained
by the least-squares regression of y on x.
• Residuals. A residual is the difference between an
observed value of the response variable and the value
predicted by the regression line. That is, residual 5 y 2 y⁄.
• Outliers and influential observations in regression. An
outlier is an observation that lies outside the overall pattern
of the other observations. Points that are outliers in the y
direction of a scatterplot have large regression residuals, but
other outliers need not have large residuals. An observation
is influential for a statistical calculation if removing it
would markedly change the result of the calculation. Points
that are outliers in the x direction of a scatterplot are often
influential for the least-squares regression line.
• Simpson’s paradox. An association or comparison
that holds for all of several groups can reverse direction
when the data are combined to form a single group. This
reversal is called Simpson’s paradox.
• Confounding. Two variables are confounded when their
effects on a response variable cannot be distinguished
from each other. The confounded variables may be either
explanatory variables or lurking variables.
Chapter 3
• Anectdotal evidence. Anecdotal evidence is based on
haphazardly selected individual cases, which often come to
our attention because they are striking in some way. These
cases need not be representative of any larger group of cases.
• Available data. Available data are data that were
produced in the past for some other purpose but that may
help answer a present question.
• Observation versus experiment. In an observational
study we observe individuals and measure variables of
interest but do not attempt to influence the responses. In
an experiment we deliberately impose some treatment on
individuals and we observe their responses.
• Experimental units, subjects, treatment. The
individuals on which the experiment is done are the
experimental units. When the units are human beings,
they are called subjects. A specific experimental condition
applied to the units is called a treatment.
• Bias. The design of a study is biased if it systematically
favors certain outcomes.
• Principles of experimental design. 1. Compare two or
more treatments. 2. Randomize—use impersonal chance
to assign experimental units to treatments. 3. Repeat each
treatment on many units to reduce chance variation in the
results.
• Statistical significance. An observed effect so large that
it would rarely occur by chance is called statistically
significant.
• Random digits. A table of random digits is a list of the
digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that has the following
properties: 1. The digit in any position in the list has the
same chance of being any one of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
2. The digits in different positions are independent in the
sense that the value of one has no influence on the value
of any other.
• Block design. A block is a group of experimental units
or subjects that are known before the experiment to
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Formulas and Key Ideas F-3
be similar in some way that is expected to affect the
response to the treatments. In a block design, the random
assignment of units to treatments is carried out separately
within each block.
• Population and sample. The entire group of individuals
that we want information about is called the population.
A sample is a part of the population that we actually
examine in order to gather information.
• Voluntary response sample. A voluntary response
sample consists of people who choose themselves by
responding to a general appeal. Voluntary response
samples are biased because people with strong opinions,
especially negative opinions, are most likely to respond.
• Simple random sample. A simple random sample (SRS)
of size n consists of n individuals from the population
chosen in such a way that every set of n individuals has an
equal chance to be the sample actually selected.
• Probability sample. A probability sample is a sample
chosen by chance. We must know what samples are
possible and what chance, or probability, each possible
sample has.
• Stratified random sample. To select a stratified random
sample, first divide the population into groups of similar
individuals, called strata. Then choose a separate SRS in
each stratum and combine these SRSs to form the full
sample.
• Undercoverage and nonresponse. Undercoverage occurs
when some groups in the population are left out of the
process of choosing the sample. Nonresponse occurs when
an individual chosen for the sample can’t be contacted or
does not cooperate.
• Basic data ethics. The organization that carries out
the study must have an institutional review board that
reviews all planned studies in advance in order to protect
the subjects from possible harm. All individuals who
are subjects in a study must give their informed consent
before data are collected. All individual data must be kept
confidential. Only statistical summaries for groups of
subjects may be made public.
Chapter 4
• Randomness and probability. We call a phenomenon
random if individual outcomes are uncertain but there
is nonetheless a regular distribution of outcomes in
a large number of repetitions. The probability of any
outcome of a random phenomenon is the proportion of
times the outcome would occur in a very long series of
repetitions.
• Sample space. The sample space S of a random
phenomenon is the set of all possible outcomes.
• Event. An event is an outcome or a set of outcomes of a
random phenomenon. That is, an event is a subset of the
sample space.
• Probability rules. Rule 1. The probability P(A) of any
event A satisfies 0 # P(A) # 1. Rule 2. If S is the sample
space in a probability model, then P(S) 5 1. Rule 3. Two
events A and B are disjoint if they have no outcomes in
common and so can never occur together. If A and B are
disjoint, P(A or B) 5 P(A) 1 P(B). This is the addition rule
for disjoint events. Rule 4. The complement of any event
A is the event that A does not occur, written as Ac. The
complement rule states that P(Ac) 5 1 2 P(A).
• Probabilities in a finite sample space. Assign
a probability to each individual outcome. These
probabilities must be numbers between 0 and 1 and
must have sum 1. The probability of any event is the
sum of the probabilities of the outcomes making up
the event.
• Equally likely outcomes. If a random phenomenon has k
possible outcomes, all equally likely, then each individual
outcome has probability 1yk. The probability of any event
A is P(A) 5 (count of outcomes in A)yk.
• The multiplication rule for independent events.
Rule 5. Two events A and B are independent if knowing
that one occurs does not change the probability
that the other occurs. If A and B are independent,
P(A and B) 5 P(A)P(B). This is the multiplication rule for
independent events.
• Random variable. A random variable is a variable whose
value is a numerical outcome of a random phenomenon.
• Discrete random variable. A discrete random variable
X has a finite number of possible values. The probability
distribution of X lists the values and their probabilities:
Value of X x1 x2 x3 Á xk
Probability p1 p2 p3 Á pk
The probabilities pi must satisfy two requirements:
1. Every probability pi is a number between 0 and 1. 2.
p1 1 p2 1 Á 1 pk 5 1. Find the probability of any event
by adding the probabilities pi of the particular values xi
that make up the event.
• Continuous random variable. A continuous random
variable X takes all values in an interval of numbers. The
probability distribution of X is described by a density curve.
The probability of any event is the area under the density
curve and above the values of X that make up the event.
• Mean and variance of a discrete random variable.
Suppose that X is a discrete random variable. The
variance of X is
2X 5 (x1 2 X)
2p1 1 (x2 2 X)
2p2 1 Á 1 (xk 2 X)2pk
To find the mean of X, multiply each possible value by its
probability, then add all the products:
X 5 x1p1 1 x2p2 1 Á 1 xkpk
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F-4 Formulas and Key Ideas
• Law of large numbers. Draw independent observations
at random from any population with finite mean . Decide
how accurately you would like to estimate . As the number
of observations drawn increases, the mean x of the observed
values eventually approaches the mean of the population
as closely as you specified and then stays that close.
• Rules for means. Rule 1. If X is a random variable and a
and b are fixed numbers, then a 1 bX 5 a 1 bX. Rule 2. If
X and Y are random variables, then X 1 Y 5 X 1 Y.
• Rules for variances and standard deviations. Rule
1. If X is a random variable and a and b are fixed
numbers, then 2a1bX 5 b
22X. Rule 2. If X and Y are
independent random variables, then 2X1Y 5
2
X 1
2
Y and
2X2Y 5
2
X 1
2
Y. This is the addition rule for variances
of independent random variables. Rule 3. If X and
Y have correlation , then 2X1Y 5
2
X 1
2
Y 1 2XY
and 2X2Y 5
2
X 1
2
Y 2 2XY. This is the general addition
rule for variances of random variables. To find the
standard deviation, take the square root of the variance.
• Rules of probability. Rule 1. 0 # P(A) # 1 for any event
A. Rule 2. P(S) 5 1. Rule 3. Addition rule: If A and B are
disjoint events, then P(A or B) 5 P(A) 1 P(B). Rule 4.
Complement rule: For any event A, P(Ac) 5 1 2 P(A). Rule
5. Multiplication rule: If A and B are independent events,
then P(A and B) 5 P(A)P(B).
• Union. The union of any collection of events is the event
that at least one of the collection occurs.
• Addition rule for disjoint events. If events
A, B, and C are disjoint in the sense that no
two have any outcomes in common, then
P(one or more of A, B, C) 5 P(A) 1 P(B) 1 P(C). This rule
extends to any number of disjoint events.
• General addition rule for unions of two events. For any
two events A and B, P(A or B) 5 P(A) 1 P(B) 2 P(A and B).
• Multiplication rule. The probability that both of
two events A and B happen together can be found by
P(A and B) 5 P(A)P(BuA). Here P(BuA) is the conditional
probability that B occurs, given the information that A
occurs.
• Definition of conditional probability. When
P(A) . 0, the conditional probability of B given A is
P(B u A) 5
P(A and B)
P(A)
.
• Intersection. The intersection of any collection of events
is the event that all of the events occur.
• Bayes’s rule. Suppose that A1, A2, ..., Ak are disjoint events
whose probabilities are not 0 and add to exactly 1. That is,
any outcome is in exactly one of these events. Then if C is
any other event whose probability is not 0 or 1,
P(Ai u C) 5
P(C u Ai)P(Ai)
P(C u A1)P(A1) 1 P(C u A2)P(A2) 1 Á 1 P(Ak)P(C u Ak)
• Independent events. Two events A and B that both have
positive probability are independent if P(BuA) 5 P(B).
Chapter 5
• Parameters and statistics. A parameter is a number that
describes the population. A parameter is a fixed number,
but in practice we do not know its value. A statistic is a
number that describes a sample (is computed from the
sample data). The value of a statistic can change from
sample to sample. We often use a statistic to estimate an
unknown parameter.
• Sampling distribution. The sampling distribution of a
statistic is the distribution of values taken by the statistic
in all possible samples of the same size from the same
population or randomized experiment.
• Bias and variability. Bias concerns the center of the
sampling distribution. A statistic used to estimate a
parameter is unbiased if the mean of its sampling
distribution is equal to the true value of the parameter
being estimated. The variability of a statistic is described
by the spread of its sampling distribution. This spread
is determined by the sampling design and the sample
size n. Statistics from larger probability samples have
smaller spreads.
• Managing bias and variability. To reduce bias, use
random sampling. When we start with a list of the entire
population, simple random sampling produces unbiased
estimates—the values of a statistic computed from an
SRS neither consistently overestimate nor consistently
underestimate the value of the population parameter. To
reduce the variability of a statistic from an SRS, use a
larger sample. You can make the variability as small as
you want by taking a large enough sample.
• Large populations do not require large samples. The
variability of a statistic from a random sample does
not depend on the size of the population, as long as the
population is at least 20 times larger than the sample.
• The sample mean x of an SRS of size n drawn from a
large population with mean and standard deviation
has a sampling distribution with mean x 5 and
standard deviation x 5 yÏn.
• Linear combinations of independent Normal random
variables have Normal distributions. In particular, if the
population has a Normal distribution, so does x.
• The central limit theorem states that for large n the
sampling distribution of x is approximately N(, yÏn)
for any population with mean and finite standard
deviation . This includes populations of both continuous
and discrete random variables.
• The binomial distribution. A count X of successes has
the binomial distribution B(n, p) when there are n trials,
all independent, each resulting in a success or a failure,
and each having the same probability p of a success.
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Formulas and Key Ideas F-5
The mean of X is X 5 np and the standard deviation is
X 5 Ïnp(1 2 p).
• The sample proportion of successes p⁄ 5 Xyn has mean
p⁄ 5 p and standard deviation p⁄ 5 Ïp(1 2 p)yn. It is an
unbiased estimator of the population proportion p.
• The sampling distribution of the count of successes.
The B(n, p) distribution is a good approximation to the
sampling distribution of the count of successes in an SRS
of size n from a large population containing proportion
p of successes. We will use this approximation when the
population is at least 20 times larger than the sample.
• The sampling distribution of the sample proportion.
The sampling distribution of p⁄ is not binomial but
the B(n, p) distribution can be used to do probability
calculations about p⁄ by restating them in terms of the
count X. We will use the B(n, p) distribution when the
population is at least 20 times larger than the sample.
• Normal approximation to the binomial distribution
says that if X is a count having the B(n, p) distribution,
then when n is large, X is approximately
N(np,Ïnp(1 2 p)). In addition, the sample proportion
p⁄ 5 Xyn is N(p,Ïp(1 2 p)yn). We will use these
approximations when np $ 10 and n(1 2 p) $ 10. The
continuity correction improves the accuracy of the
Normal approximations.
• The binomial probability formula is
P(X 5 k) 5 Snk2 pk(1 2 p)n2k
where the possible values of X are k 5 0, 1, Á , n. The
binomial probability formula uses the binomial coefficient
1nk2 5 n!k! (n 2 k)!
Here the factorial n! is
n! 5 n 3 (n 2 1) 3 (n 2 2) 3 Á 3 3 3 2 3 1
for positive whole numbers n and 0! 5 1. The binomial
coefficient counts the number of ways of distributing k
successes among n trials.
• The Poisson distribution. A count X of successes has a
Poisson distribution in the Poisson setting: the number
of successes that occur in two nonoverlapping units of
measure are independent; the probability that a success
will occur in a unit of measure is the same for all units of
equal size and is proportional to the size of the unit; the
probability that more than one event occurs in a unit of
measure is negligible for very small-sized units. In other
words, the events occur one at a time. The mean of X is
and the standard deviation of X is Ï.
• The Poisson probability that X takes any of the whole
numbers 0, 1, 2, 3, and so on is
P(X 5 k) 5
e2k
k!
k 5 0, 1, 2, 3, Á
• Sums of independent Poisson random variables also
have the Poisson distribution. For example, in a Poisson
model with mean per unit of measure, the count of
successes in a units is a Poisson random variable with
mean a.
Chapter 6
• Confidence interval. The purpose of a confidence
interval is to estimate an unknown parameter with an
indication of how accurate the estimate is and of how
confident we are that the result is correct. Any confidence
interval has two parts: an interval computed from the data
and a confidence level. The interval often has the form
estimate 6 margin of error.
• Margin of error. The margin of error is obtained from the
sampling distribution and indicates how much error can
be expected because of chance variation.
• Confidence level. The confidence level states the
probability that the method will give a correct answer.
That is, if you use 95% confidence intervals, in the long
run 95% of your intervals will contain the true parameter
value. When you apply the method once, you do not know
if your interval gave a correct value (this happens 95% of
the time) or not (this happens 5% of the time).
• Confidence interval for the mean . For a Normal
population with known standard deviation , a level C
confidence for the mean is given by x 6 m, where the
margin of error m 5 z* Ïn. Here z
* is obtained from the
standard Normal distribution such that the probability is
C that a standard Normal random variable takes a value
between 2z* and z*.
• How confidence intervals behave. Other things being
equal, the margin of error of a confidence interval
decreases as the confidence level C decreases, the sample
size n increases, and the population standard deviation
decreases. The sample size n required to obtain a
confidence interval of specified margin of error m for a
Normal mean is n 5 (z*ym)2, where z* is the critical point
for the desired level of confidence.
• A test of significance is intended to assess the evidence
provided by data against a null hypothesis H0 in favor of
an alternative hypothesis Ha. The hypotheses are stated in
terms of population parameters. Usually H0 is a statement
that no effect or no difference is present, and Ha says that
there is an effect or difference. The difference can be in
a specific direction (one-sided alternative) or in either
direction (two-sided alternative).
• The test statistic and P-value. The test of significance
is based on a test statistic. The P-value is the probability,
computed assuming that H0 is true, that the test statistic
will take a value at least as extreme as that actually
observed. Small P-values indicate strong evidence against
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F-6 Formulas and Key Ideas
H0. Calculating P-values requires knowledge of the
sampling distribution of the test statistic when H0 is true.
If the P-value is as small or smaller than a specified value ,
the data are statistically significant at significance level .
• Significance test concerning an unknown mean m.
Significance tests for the hypothesis H0: 5 0 are based
on the z statistic, z 5 (x 2 0)y(yÏn). This z test assumes
an SRS of size n, known population standard deviation ,
and either a Normal population or a large sample.
• Power. The power of a significance test measures
its ability to detect an alternative hypothesis. The
power to detect a specific alternative is calculated as
the probability that the test will reject H0 when that
alternative is true. This calculation requires knowledge
of the sampling distribution of the test statistic under
the alternative hypothesis. Increasing the size of the
sample increases the power when the significance level
remains fixed.
• Type I and Type II errors. In the case of testing H0
versus Ha, decision analysis chooses a decision rule on the
basis of the probabilities of two types of error. A Type I
error occurs if H0 is rejected when it is in fact true. A Type
II error occurs if H0 is accepted when in fact Ha is true. In
a fixed level significance test, the significance level is
the probability of a Type I error, and the power to detect a
specific alternative is 1 minus the probability of a Type II
error for that alternative.
Chapter 7
• Standard error. When the standard deviation of a
statistic is estimated from the data, the result is called the
standard error of the statistic. The standard error of the
sample mean x is SEx 5
s
Ïn.
• The t distributions. Suppose that an SRS of size n is
drawn from an N(, ) population. The one-sample t
statistic t 5 (x 2 )y(syÏn) has the t distribution with
n 2 1 degrees of freedom.
• The one-sample t confidence interval. Consider an
SRS of size n drawn from a population having unknown
mean . A level C confidence interval for is x 6 t*syÏn,
where t* is the value for the t(n 2 1) density curve with
area C between 2t* and t*. The quantity t*syÏn is the
margin of error. This interval is exact when the population
distribution is Normal and is approximately correct for
large n in other cases.
• The one-sample t test. Suppose that an SRS of size n
is drawn from a population having unknown mean . To
test the hypothesis H0: 5 0, compute the one-sample t
statistic t 5 (x 2 0)y(syÏn). P-values or fixed significance
levels are computed from the t(n 2 1) distribution.
• Matched pairs t procedures. These procedures are
needed when subjects or experimental units are matched
in pairs or when there are two measurements on each
individual or experimental unit and the question of interest
concerns the difference between the two measurements.
These one-sample procedures involve first taking the
differences within each matched pair to produce a single
sample.
• One-sample equivalence testing assesses whether
a population mean is practically different from a
hypothesized mean 0. This test requires a threshold ,
which represents the largest difference between and 0
such that the means are considered equivalent.
• Robustness of the one-sample t procedures. The t
procedures are relatively robust against non-Normal
populations. The t procedures are useful for non-Normal
data when 15 # n , 40 unless the data show outliers or
strong skewness. When n $ 40, the t procedures can be
used even for clearly skewed distributions.
• The two-sample t test. Suppose that an SRS of size n1
is drawn from a Normal population with unknown mean
1 and that an independent SRS of size n2 is drawn from
another Normal population with unknown mean 2. To
test the hypothesis H0: 1 5 2, compute the two-sample t
statistic t 5 (x1 2 x2)y(Ïs21yn1 1 s22yn2) and use P-values
or critical values for the t(k) distribution, where the
degrees of freedom k are either approximated by software
or are the smaller of n1 2 1 and n2 2 1.
• The two-sample t confidence interval. Suppose that an
SRS of size n1 is drawn from a Normal population with
unknown mean m1 and that an independent SRS of size n2
is drawn from another Normal population with unknown
mean m2. The confidence interval for m1 2 m2 is given by
(x1 2 x2) 6 t
*Ïs21yn1 1 s22yn2. This interval has confidence
level at least C no matter what the population standard
deviations may be. Here, t* is the value for the t(k) density
curve with area C between 2t* and t*, where the degrees
of freedom k are either approximated by software or are
the smaller of n1 2 1 and n2 2 1.
• Robustness of the two-sample t procedures. The
guidelines for practical use of two-sample t procedures
are similar to those for one-sample t procedures using the
sum of the sample sizes n1 1 n2 for n. Equal sample sizes
are recommended.
• Pooled two-sample t procedures. If we can
assume that the two populations have equal
variances, pooled two-sample t procedures can
be used. These are based on the pooled estimator
s2p 5 ((n1 2 1)s
2
1 1 (n2 2 1)s
2
2)y(n1 1 n2 2 2) of the
unknown common variance and the t(n1 1 n2 2 2)
distribution.
• Sample size needed for an expected margin of error.
The sample size required to obtain a confidence interval
with an expected margin of error no larger than m for a
population mean satisfies the constraint
m $ t*s*yÏn
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Formulas and Key Ideas F-7
where t* is the critical value for the desired level of
confidence with n 2 1 degrees of freedom, and s* is the
guessed value for the population standard deviation.
The sample sizes necessary for a two-sample confidence
interval can be obtained using a similar constraint, but
guesses of both standard deviations and an estimate for
the degrees of freedom are required. To be conservative
use the smaller of n1 2 1 and n2 2 1.
• Power of a t test. The power of the one-sample t test is
calculated like that of the z test, using an approximate
value for both s and s. The power of the two-sample
t test is found by first finding the critical value for
the significance test, the degrees of freedom, and the
noncentrality parameter for the alternative of interest.
These are used to calculate the power from a noncentral
t distribution. Software can often be used to do these
calculations.
• Sign test. The sign test is a distribution-free test because
it uses probability calculations that are correct for a wide
range of population distributions. The sign test for “no
treatment effect” in matched pairs counts the number of
positive differences. The P-value is computed from the
B(n, 1y2) distribution, where n is the number of non-0
differences. The sign test is less powerful than the t test
in cases where use of the t test is justified.
Chapter 8
• Large-sample confidence interval for a population
proportion. Choose an SRS of size n from a large
population with an unknown proportion p of
successes. The sample proportion is p⁄ 5 Xyn, where
X is the number of successes. The standard error of p⁄
is SEp⁄ 5 Ï(p⁄ (1 2 p⁄ )yn and the margin of error for
confidence level0 C is m 5 z*SEp⁄, where the critical
value z* is the value for the standard Normal density
curve with area C between 2z* and z*. An approximate
level C confidence interval for p is p⁄ 6 m. Use this
interval for 90%, 95%, or 99% confidence when the
number of successes and the number of failures are
both at least 10.
• Large-sample significance test for a population
proportion. Draw an SRS of size n from a large
population with an unknown proportion p of successes.
To test the hypothesis H0: p 5 p0, compute the z
statistic, z 5 (p⁄ 2 p0)yÏ(p0(1 2 p0)yn. In terms of a
standard Normal random variable Z, the approximate
P-value for a test of H0 against Ha: p . p0 is P(Z $ z),
Ha: p , p0 is P(Z # z), and Ha: p Þ p0 is 2P(Z $ uzu).
• Sample size for desired margin of error for a single
proportion. The level C confidence interval for a
proportion p will have a margin of error approximately
equal to a specified value m when the sample size
satisfies n 5 (z*ym)2p*(1 2 p*). Here z* is the critical
value for confidence C, and p* is a guessed value for the
proportion of successes in the future sample. The margin
of error will be less than or equal to m if p* is chosen
to be 0.5. The sample size required when p* 5 0.5 is
n 5 (1y4)(z*ym)2.
• Sample size for a significance test for a single
proportion. Statistical software can be used to
determine the sample size needed for a significance test
to compare two proportions. Inputs required are the
null and alternative hypotheses, the type I error, and the
desired power.
• Large-sample confidence interval for comparing
two proportions. Choose an SRS of size n1 from a
large population having proportion p1 of successes
and an independent SRS of size n2 from another
population having proportion p2 of successes.
The estimate of the difference in the population
proportions is D 5 p⁄ 1 2 p
⁄
2. The standard error of D
is SED 5 Ï(p⁄ 1(1 2 p⁄ 1)yn1) 1 (p⁄ 2(1 2 p⁄ 2)yn2) and the
margin of error for confidence level C is m 5 z*SED,
where the critical value z* is the value for the standard
Normal density curve with area C between 2z* and
z*. An approximate level C confidence interval for
p1 2 p2 is D 6 m. Use this method for 90%, 95%, or
99% confidence when the number of successes and the
number of failures in each sample are at least 10.
• Significance test for comparing two proportions.
To test the hypothesis H0: p1 5 p2 compute the z
statistic z 5 (p⁄ 1 2 p
⁄
2)ySEDp), where the pooled standard
error is SEDp 5 Ï(p⁄ (1 2 p⁄ )(1yn1 1 1yn2) and where
p⁄ 5 (X1 1 X2)y(n1 1 n2). In terms of a standard Normal
random variable Z, the P-value for a test of H0 against
Ha: p1 . p2 is P(Z $ z), Ha: p1 , p2 is P(Z # z), and
Ha: p1 Þ p2 is 2P(Z $ uzu).
• Sample size for desired margin of error for the
difference between two proportions. The level C
confidence interval for the difference between two
proportions will have a margin of error approximately
equal to a specified value m when the sample size satisfies
n 5 1z
*
m2
2
(p*1(1 2 p*1) 1 p*2(1 2 p*2))
Here z* is the critical value for confidence C, and p*1 and
p*2 are guessed values for p1 and p2, the proportions of
successes in the future sample. The margin of error will
be less than or equal to m if p*1 and p*2 are chosen to be
0.5. The common sample size required is then given by
n 5 11221z
*
m2
2
• Sample size for a significance test to compare
two proportions. Statistical software can be used to
determine the sample size needed for a significance test
to compare two proportions. Inputs required are the
null and alternative hypotheses, the Type I error, and the
desired power.
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F-8 Formulas and Key Ideas
Chapter 9
• Chi-square statistic. The chi-square statistic is a
measure of how much the observed cell counts in a two-
way table diverge from the expected cell counts. The
formula for the statistic is
X2 5 o
(observed count 2 expected count)2
expected count
where “observed” represents an observed cell count,
“expected” represents the expected count for the same
cell, and the sum is over all r 3 c cells in the table.
• Chi-square test for two-way tables. The null
hypothesis H0 is that there is no association between
the row and column variables in a two-way table. The
alternative is that these variables are related. If H0 is
true, the chi-square statistic X2 has approximately a 2
distribution with (r 2 1)(c 2 1) degrees of freedom. The
P-value for the chi-square test is P(2 $ X2), where 2
is a random variable having the 2(df) distribution with
df 5 (r 2 1)(c 2 1).
• Expected cell counts.
expected count 5 (row total 3 column total)yn.
• The chi-square goodness-of-fit test. Data for n
observations of a categorical variable with k possible
outcomes are summarized as observed counts,
n1, n2, . . ., nk in k cells. A null hypothesis specifies
probabilities p1, p2, . . ., pk for the possible outcomes.
For each cell, multiply the total number of observations
n by the specified probability to determine the expected
counts: expected count 5 npi. The chi-square statistic
measures how much the observed cell counts differ from
the expected cell counts. The formula for the statistic is
X2 5 o
(observed count 2 expected count)2
expected count
The degrees of freedom are k 2 1, and P-values are
computed from the chi-square distribution.
Chapter 10
• Simple linear regression. The statistical model for
simple linear regression assumes that the means of the
response variable y fall on a line when plotted against
x, with the observed y’s varying Normally about these
means. For n observations, this model can be written
yi 5 b0 1 b1xi 1 i, where i 5 1, 2, Á , n, and the i are
assumed to be independent and Normally distributed
with mean 0 and standard deviation s. Here b0 1 b1xi is
the mean response when x 5 xi. The parameters of the
model are b0, b1, and s.
• Estimation of model parameters. The population
regression line intercept and slope, b0 and b1, are
estimated by the intercept and slope of the least-squares
regression line, b0 and b1. The parameter s is estimated
by s 5 Ïoe2i y(n 2 2), where the ei are the residuals
ei 5 yi 2 y
⁄
i.
• Confidence interval and significance test for b1.
A level C confidence interval for population slope
b1 is b1 6 t
*SEb1 where t
* is the value for the t(n 2 2)
density curve with area C between 2t* and t*. The test
of the hypothesis H0: b1 5 0 is based on the t statistic
t 5 b1ySEb1 and the t(n 2 2) distribution. This tests
whether there is a straight-line relationship between
y and x. There are similar formulas for confidence
intervals and tests for 0, but these are meaningful only
in special cases.
• Confidence interval for the mean response.
The estimated mean response for the subpopulation
corresponding to the value x* of the explanatory
variable is m⁄ y 5 b0 1 b1x
*. A level C confidence
interval for the mean response is m⁄ y 6 t*SEm⁄ where t
*
is the value for the t(n 2 2) density curve with area C
between 2t* and t*.
• Prediction interval for the estimated response. The
estimated value of the response variable y for a future
observation from the subpopulation corresponding to
the value x* of the explanatory variable is y⁄ 5 b0 1 b1x*.
A level C prediction interval for the estimated response
is y⁄ 6 t*SEy⁄ where t
* is the value for the t(n 2 2)
density curve with area C between 2t* and t*. The
standard error for the prediction interval is larger
than the confidence interval because it also includes
the variability of the future observation around its
subpopulation mean.
Chapter 11
• Multiple linear regression. The statistical model
for multiple linear regression with response
variable y and p explanatory variables x1, x2, Á , xp
is yi 5 b0 1 b1xi1 1 b2xi2 1 Á 1 bpxip 1 i where
i 5 1, 2, Á , n. The i are assumed to be independent
and Normally distributed with mean 0 and standard
deviation . The parameters of the model are
b0 , b1 , b2, Á , bp, and s.
• Estimation of model parameters. The multiple
regression equation predicts the response variable by
a linear relationship with all the explanatory variables:
y⁄ 5 b0 1 b1x1 1 b2x2 1 Á 1 bpxp. The b’s are estimated
by b0, b1, b2, Á , bp, which are obtained by the method
of least squares. The parameter s is estimated by
s 5 ÏMSE 5 ÏSe2i y(n 2 p 2 1) where the ei are the
residuals, ei 5 yi 2 y
⁄
i.
• Confidence interval for bj. A level C confidence
interval for bj is bj 6 t
*SEbj where t
* is the value for the
t(n 2 p 2 1) density curve with area C between 2t* and
t*. The test of the hypothesis H0: bj 5 0 is based on the t
statistic t 5 bjySEbj and the t(n 2 p 2 1) distribution. The
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Formulas and Key Ideas F-9
estimate bj of bj and the test and confidence interval for
bj are all based on a specific multiple linear regression
model. The results of all of these procedures change if
other explanatory variables are added to or deleted from
the model.
• The ANOVA F test. The ANOVA table for a multiple
linear regression gives the degrees of freedom, sum of
squares, and mean squares for the model, error, and
total sources of variation. The ANOVA F statistic is the
ratio MSM/MSE and is used to test the null hypothesis
H0: b1 5 b2 5 Á 5 bp 5 0. If H0 is true, this statistic has
an F(p, n 2 p 2 1) distribution.
• Squared multiple correlation. The squared multiple
correlation is given by the expression R2 5 SSMySST and
is interpreted as the proportion of the variability in the
response variable y that is explained by the explanatory
variables x1, x2, Á , xp in the multiple linear regression.
Chapter 12
• One-way analysis of variance (ANOVA) is used to
compare several population means based on independent
SRSs from each population. The populations are assumed
to be Normal with possibly different means and the same
standard deviation. To do an analysis of variance, first
compute sample means and standard deviations for all
groups. Side-by-side boxplots give an overview of the
data. Examine Normal quantile plots (either for each
group separately or for the residuals) to detect outliers or
extreme deviations from Normality. Compute the ratio of
the largest to the smallest sample standard deviation. If
this ratio is less than 2 and the Normal quantile plots are
satisfactory, ANOVA can be performed.
• ANOVA F test. An analysis of variance table organizes
the ANOVA calculations. Degrees of freedom, sums of
squares, and mean squares appear in the table. The F
statistic is the ratio MSG/MSE and is used to test the null
hypothesis that the population means are all equal. The
alternative hypothesis is true if there are any differences
among the population means. The F(I 2 1, N 2 I)
distribution is used to compute the P-value.
• Contrasts. Specific questions formulated before
examination of the data can be expressed as contrasts.
A contrast is a combination of population means of
the form c 5 oaimi where the coefficients ai sum to
0. The corresponding sample contrast is c 5 oaixi.
The standard error of c is SEc 5 spÏoa2i yni. Tests and
confidence intervals for contrasts provide answers to
these specific questions.
• Multiple comparisons. To perform a multiple-
comparisons procedure, compute t statistics for all pairs of
means using the formula tij 5 (xi 2 xj)y(spÏ1yni 1 1ynj).
If utiju $ t** we declare that the population means mi and mj
are different. Otherwise, we conclude that the data do not
distinguish between them. The value of t** depends upon
which multiple-comparisons procedure we choose.
Chapter 13
• Two-way analysis of variance is used to compare
population means when populations are classified
according to two factors. ANOVA assumes that the
populations are Normal with possibly different means
and the same standard deviation and that independent
SRSs are drawn from each population. As with one-
way ANOVA, preliminary analysis includes examination
of means, standard deviations, and Normal quantile
plots.
• ANOVA table and F tests. ANOVA separates the
total variation into parts for the model and error. The
model variation is separated into parts for each of the
main effects and the interaction. These calculations
are organized into an ANOVA table. Pooling is used
to estimate the within-group variance. F statistics and
P-values are used to test hypotheses about the main
effects and the interaction.
• Marginal means are calculated by taking averages of the
cell means across rows and columns. Careful inspection
of the cell means is necessary to interpret statistically
significant main effects and interactions. Plots are a
useful aid.
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