Transcript AP Statistics Review - William H. Peacock, LCDR USN, Ret

AP Statistics Review
Part IV & Part V & Part VI & Part VII:
Randomness and Probability
From the Data at Hand to the World at Large
Learning About the World
Inference When Variables Are Related
Part IV:
Chapters 14 - 17
RANDOMNESS AND
PROBABILITY
PROBABILITY
Probability
The probability of an event is its long-run
relative frequency.
Vocabulary
• Trial – a single attempt or realization of a
random phenomenon.
• Outcome – is the value measured,
observed, or reported for each trial.
• Sample Space – the set of all possible
outcomes.
• Event – a collection of outcomes. Usually
designated by capital letters.
• Independence – Two events are
independent if the occurrence of one event
does not alter the probability that the other
event occurs.
Vocabulary
• Law of Large Numbers – The long-run
relative frequency of repeated
independent events gets closer and closer
to the true relative frequency as the
number of trials increases.
Probability Rules
• The probability of an event is a number from 0 to 1
that reports the likelihood of the event’s occurrence.
0≤P(A)≤1
• Theoretical Probability of an Event A (when all
possible outcomes are equally likely):
P ( A) 
number of elements in A
number of elements in the sample space (S)
• Empirical (Experimental) Probability of an Event A:
P( A) 
number of successes
total number of observations
An experimental probability arrived at by simulation will
probably not be exactly equal to the corresponding
theoretical value, but it should approach this value in the
long run.
Probability Rules (cont.)
• The probability of the set of all possible
outcomes of a trial (sample space) must
equal 1. P(S)=1
• The Complement of an Event A: (Ac, A’)
is the set of all possible outcomes that are
not in event A.
• The probability of the complement of an
event is 1 minus the probability that the
event occurs. P(Ac)=1-P(A)
Probability Rules (cont.)
• Addition Rule for Disjoint (Mutually
Exclusive) Events: Two events, A and B,
are disjoint if they cannot occur together.
The probability that one or the other or
both occur is the sum of the probabilities of
the two events.
P(A or B)=P(A⋃B)=P(A)+P(B)
• General Addition Rule:
P(A or B)=P(A⋃B)=P(A)+P(B)-P(A⋂B)
Probability Rules (cont.)
• Multiplication Rule for Independent
Events: For two independent events A and
B, the probability that both A and B occur
is the product of the two probabilities.
P(A and B)=P(A⋂B)=P(A)⤫P(B)
• Conditional Probability and the General
Multiplication Rule: When the events are
not independent.
• Conditional Probability: When the
probability of the second event is
conditioned upon the first event having
occurred. Notation P(B|A)
P( A  B)
P ( B | A) 
P ( A)
Probability Rules (cont.)
• General Multiplication Rule: The
probability that events A and B both occur
is the probability that A occurs times the
probability that B occurs given that A has
already occurred.
P(A and B)=P(AB)=P(A)⤫P(B|A)
• Formal Definition of Independence: Two
events are independent if the outcome of
one event does not influence the
probability of the other. P(B|A)=P(B)
• Disjoint events cannot be independent.
RANDOM VARIABLES
Random Variables
• A random variable assumes any of several
different values as a result of some
random event.
• Random variables are denoted by capital
letters such as X, and the values that the
random variable can take on are denoted
by the same lowercase variable (x1, x2,…).
• All possible values that the random
variable can assume, with their associated
probabilities, form the probability
distribution for the random variable.
Types of Random Variables
• Continuous Random Variable – Is a random
variable that can take on any numeric value
within a range of values.
• The probability distribution of a continuous
random variable is described by a density
curve (mathematical model).
• The probability of any given event is the area
under the curve over the values of x that make
up the event.
• A density curve lies above the x-axis and has
an area beneath it of 1.
• Discrete Random Variable – is a random
variable that can take on a finite number of
distinct outcomes.
Discrete Random Variable
• Expected Value – The theoretical long-run
average of a random value, a weighted
average. μ=E(X)=ΣxP(X)
• Standard Deviation –
Var ( X )   2    x    P( x)
2
SD( X )    Var ( X )
When calculating the expected winnings for a
game of chance, don’t forget to take into
account the cost to play the game.
Operations on Random Variables
Let X be a random variable with mean μx and
variance σ2x and let Y be a new random
variable such that Y=a±bx. The mean
(expected value) and variance of this new
random variable are given by the following.
Y  ( a  bX )  a  b X
 Y2   (2a  bX )  b 2 X2
and  Y   ( a  bX )  b 2 X2  b X
More About Means and
Variances
• The mean of the sum of two random
variables is the sum of the means of the
random variables.
• The mean of the difference of two random
variables is the difference of the means of
the two random variables.
μX±Y=E(X±Y)=E(X)±E(Y)
• If the random variables are independent,
the variance of their sum or difference is
always the sum of the variances.
σ2X±Y=Var(X±Y)=Var(X)+Var(Y)
Variances of independent variables add;
Standard Deviations DO NOT ADD!
PROBABILITY MODELS
Geometric Model
• Each observation falls into one of two
settings: success or failure.
• The probability of success, p, is the same
for each observation.
• The observations are independent.
• The variable of interest is the number of
trials required to obtain the first success.
Geometric Model
Binomial Model
• Each observation falls into one of two
settings: success or failure.
• There is a fixed number n of observations.
• The n observations are all independent.
• The probability of success, p, is the same
for each observation.
Binomial Model
Geometric vs. Binomial Models
• The difference in conditions between the
binomial and geometric models is in the
number of trials.
• The binomial requires a fixed number of
trials to be set in advance.
• The geometric has no fixed number of
trials and is often used to model wait-time.
• In other words, we continue conducting
trials until we get our first success.
Normal Model as an
Approximation to the Binomial
• The Normal distribution can be used as an
approximation for the binomial distribution
provided that the number of successes
and failures is at least ten (np≥10 and
nq≥10).
• It is convenient to use this method of
approximation when the number of trials is
large and beyond the ability of the
calculator to compute probabilities.
What You need to Know
• Law of Large Numbers.
• Terminology: trial, outcome, event, sample
space.
• Disjoint/Mutually Exclusive
• Independence
• Note: Disjoint events CANNOT be
independent!
• Valid Probabilities
• Complement Rule
• Building a Venn Diagram, a Table, or a Tree.
• Using whatever method to determine
probabilities, unions, intersections, and
conditionals.
What You need to Know
• How to construct a probability table.
• How to find the mean, variance, and standard
deviation of a probability distribution.
• How adding and subtracting affect the mean
and standard deviation.
• How multiplying affects the mean and
standard deviation.
• How to combine means and standard
deviations (Remember.. VARIANCES
ALWAYS ADD!).
• How to identify a binomial or geometric
situation.
• How to find binomial or geometric probabilities.
• How to find the mean and standard deviation
for binomial and geometric situations.
PRACTICE PROBLEMS
#1
If P(A) = 0.5, P(B) = 0.6, and P(A  B) =
0.3, then P(A  B) is
a)
b)
c)
d)
0.8
0.5
0.6
0
#2
If P(A) = 0.5, P(B) = 0.4, and P(B|A) =
0.3, then P(A  B) is
a)
b)
c)
d)
0.75
0.6
0.12
0.15
#3
If A and B are mutually excusive events
and P(A) = 0.2 and P(B) = 0.7, then
P (A  B) is
a)
b)
c)
d)
0.14
0
0.9
0.2857
#4
In a particular rural region, 65% of the
residents are smokers, and research
indicates that 15 percent of the smokers
have some form of lung cancer. The
probability of a resident is a smoker and
has lung cancer is
a)
b)
c)
d)
0.0975
0.2308
0.15
0.65
#5
In a 1974 “Dear Abby” letter, a woman
lamented that she had just given birth to her
eighth child and all were girls! Her doctor
had assured her that the chance of an 8th
girl was only 1 in 100. What was the real
probability that the eighth child would be a
girl? Before the birth of the first child, what
was the probability that the woman would
have eight girls in a row?
a)
b)
c)
d)
e)
.5, .0039
.0039, .0039
.5, .5
.0039, .4
.5, .01
#6
A randomly selected student is asked to
respond to yes, no, or maybe to the
question, “Do you intend to vote in the
next presidential election?” The sample
space is {yes, no, maybe}. Which of the
following represents a legitimate
assignment of probabilities for this
sample space?
a)
b)
c)
d)
e)
.4, .4, .2
.4, .6, .4
.3, .3, .3
.5, .3, -.2
None of the above
#7
If you choose a card at random from a
well-shuffled deck of 52 cards, what is
the probability that the card chosen is
not a heart?
a)
b)
c)
d)
.25
.50
.75
1
#8
You play tennis regularly with a friend,
and from past experience, you believe
that the outcome of each match is
independent. For any given match you
have a probability of 0.6 of winning. The
probability that you win the next two
matches is
a)
b)
c)
d)
e)
0.16
0.36
0.4
0.6
1.2
#9
Suppose that, in a certain part of the
world, in any 50 year period, the
probability of a major plague is 0.39, the
probability of a major famine is 0.52, and
the probability of both a plague and a
famine is 0.15. What is the probability of
a famine given that there is a plague?
a)
b)
c)
d)
e)
0.24
0.288
0.37
0.385
0.76
#10
There are two games involving flipping a coin.
In the first game you win a prize if you can
throw between 40% and 60% heads. In the
second game you win if you can throw more
than 75% heads. For each game would you
rather flip the coin 50 times or 500 times?
a) 50 times for each game.
b) 500 times for each game
c) 50 times for the first game, 500 for the
second
d) 500 times for the first game, 50 for the
second
e) The outcomes of the games do not depend
on the number of flips.
#11
Of the coral reef species on the Green
Barrier Reef off Australia, 73% are
poisonous. If a tourist boat taking divers
to different points off the reef encounters
an average of 25 coral reef species,
what are the mean and standard
deviation for the expected number of
poisonous species seen?
a.
b.
c.
d.
e.
Mu=6.75, sigma=4.93
Mu=18.25, sigma=2.22
Mu=18.25, sigma=4.93
Mu=18.25, sigma=8.88
None of the above
#12
According to a CBS/New York Times
poll taken in 1992, 15% of the public
have responded to a telephone call-in
poll. In a random group of five people,
what is the probability that exactly two
have responded to a call-in poll.
a.
b.
c.
d.
e.
.138
.165
.300
.835
.973
#13
A television game show has three
payoffs. Sixty percent of the players win
nothing. 30% of the player win $1000.
The remaining 10% win $10,000. What
are the mean and standard deviation of
the winnings?
a.
b.
c.
d.
e.
Mean=1300, st. dev=2934
Mean=1300, st. dev=8802
Mean=3667, st. dev=4497
Mean=3667, st. dev=5508
None of the above
#14
An inspection procedure at a
manufacturing plant involves picking
three items at random and then
accepting the whole lot if at least two of
the three items are in perfect condition.
If in reality, 90% of the whole lot are
perfect, what is the probability that the
lot will be accepted?
a.
b.
c.
d.
e.
.600
.667
.729
.810
.972
#15
At a warehouse sale 100 customers are
invited to choose one of 100 identical
boxes. Five boxes contain $700 color
TV sets, 25 boxes contain $540
camcorders, and the remaining boxes
contain $260 cameras. What should a
customer be willing to pay to participate
in the sale?
a.
b.
c.
d.
e.
$260
$352
$500
$540
$699
#16
The average annual incomes of high
school and college graduates in a midwestern town are $21,000 and $35,000,
respectively. If a company hires only
personnel with at least a high school
diploma and 20% of its employees have
been through college, what is the mean
income of the employees?
a.
b.
c.
d.
e.
$23,800
$27,110
$28,000
$32,200
$56,000
#17
Suppose the average height of
policemen is 71 inches with a standard
deviation of 4 inches, while the average
for policewomen is 66 inches with
standard deviation of 3 inches. What is
the mean and standard deviation for the
difference in heights between the
policemen and policewomen?
a.
b.
c.
d.
e.
Mean=5, st. dev=1
Mean=5, st. dev=3.5
Mean=5, st. dev=5
Mean=68.5, st. dev=1
Mean=68.5, st. dev=3.5
Part V:
Chapters 18 - 22
FROM THE DATA AT HAND TO
THE WORLD AT LARGE
SAMPLING DISTRIBUTION
MODELS
Sampling Distributions
• A sampling distribution is the distribution of a
statistic, such as x-bar or p-hat, from all
possible samples of a given size (n) from a
population of size N.
• A certain proportion (p) of the population may
display a particular characteristic. In each
randomly selected sample of sze n, that same
characteristic will be seen in a proportion (phat) of the sample.
• We use p-hat (our sample proportion) as an
estimate of p (the population proportion).
• The same is true for the sample mean (x-bar)
as an estimate of the population mean (μ).
Modeling the Distribution of
Sample Proportions
The sampling distribution of sample proportions has these characteristics.
• Shape – The distribution of sample proportions is approximately normally
distributed provided that
• np≥10 (there are at least 10 successes) and
• n(1-p)≥10 (there are at least 10 failures)
• Center – The mean of the sampling distribution of sample proportions,
(p-hat) is centered at the mean of the population, p, regardless of sample
size. (We say that p-hat is an unbiased estimator of p).
• Spread – The standard deviation of sampling distribution of sample
proportions is given by the formula  pˆ 
p(1  p)
, provided that
n
• 10n≥N (since we are sampling without replacement, the sample size
should be less than 10% of the population size in order to use this
formula for standard error).
Modeling the Distribution of
Sample Means
• For categorical variables, we are interested in a proportion (p). For
quantitative variables, a parameter of interest is the average or
mean (μ).
• The sampling distribution of sample means has these
characteristics.
• Shape – As sample size increases, the distribution of sample
means tends toward an approximately normal distribution (Central
Limit Theorem).
• Center – The mean of the sampling distribution of sample means,
 x , is centered at the mean of the population, μ, regardless of
sample size (We say that x-bar is an unbiased estimator of μ).
• Spread – The standard deviation of the sampling distribution of
sample means is given by the formula    .
x
n
Central Limit Theorem
• The Central Limit Theorem (CLT) says that
means of repeated samples will tend to follow
a normal model if the sample size is “large
enough”; this true no matter what shape the
population distribution has!
• If the population is not Normal, the sample size
should be at least 30 for the CLT to apply.
• Avoid confusing the distribution of a sample
with the sampling distribution model for sample
means. The distribution of a (one) sample will,
as n increases, look more like the population
from which it was taken.
Standard Error
• For both sampling distribution models
(proportions and means), we have
assumed that we know the population
parameters ρ or μ and σ.
• In most cases, it is not possible to know
these values. Consequently, we must use
what we do know and estimate the
standard deviations of sampling
distributions.
• When this happens, we call the estimated
standard deviation the standard error.
Standard Error (cont.)
• For a proportion, the standard error of p-hat is:
SE ( pˆ )  s pˆ 
pˆ 1  pˆ 
n
• For a sample mean, the standard error of x-bar is:
SE ( x )  sx 
s
n
• We estimate a population parameter, p, by using a sample
statistic, pˆ . For this reason, pˆ is called a point estimate for p.
• In general, if the expected value of the estimator equals the
corresponding parameter, the statistic is called an unbiased
estimator.
• Since  pˆ  p, pˆ is an unbiased estimator for p. (Similarly, x is
a point estimate for μ, and since  x   , x is an unbiased
estimator for μ.)
Standard Error (cont.)
• We have seen that sample statistics vary. So rather than
simply giving a single value for our estimate of the population
proportion, we use our sample estimate, pˆ , to build a range
of plausible values.
• To do this, we want to establish an interval that we believe
(at least with some degree of certainty) contains the true
value of our population parameter.
• This interval forms an interval estimate for the parameter, p,
and is called a confidence interval.
• Because of sampling variability, we can never say we are
100% certain; therefore, the confidence level we choose
indicates the degree of certainty we have that our interval
captures the true value of the parameter.
CONFIDENCE INTERVALS FOR
PROPORTIONS
Creating and Interpreting a
Confidence Interval
• What We Say
• If we choose 95% as our confidence
level, we build an interval that stretches
about 2 SE’s from pˆ in either direction.
• Then we say: We are 95% confident
that the true proportion is between
(calculated lower limit) and (calculated
upper limit).
• This is a correct interpretation of the
confidence interval.
Creating and Interpreting a
Confidence Interval
• It is also possible to interpret the
confidence level.
• A correct interpretation of the confidence
level would be: 95% confidence means
that, in the long run, 95 out of 100 intervals
calculated using the same procedure
would (using random samples of the same
size from the same population) capture the
true population proportion.
• Commonly used values for confidence
levels are 90%, 95%, and 99%, but other
levels may be chosen.
Creating and Interpreting a
Confidence Interval
Know the difference between confidence
interval and confidence level and be sure to
give the correct interpretation. It’s easy to get
interval and level confused.
Creating and Interpreting a
Confidence Interval
• The general form of all confidence
intervals is:
Estimate ± (Critical Value)(Standard Error)
• The desired confidence level determines
the critical value (the number of standard
deviations on either side of the estimate).
The critical value (CV) is denoted by z*.
• The most commonly used values are:
Level
z*
90%
1.645
95%
1.960
99%
2.576
Conditions for a One-Proportion
z-Interval
• Randomization Condition – proper
randomization techniques used to collect
the data.
• 10% Condition – sample size is less than
10% of the population.
• Success/Failure Condition – have at
least 10 “successes”  npˆ  10  and 10
“failures”  n 1  pˆ   10 .
Margin of Error
• The distance of the endpoints of the interval
from pˆ is called the margin of error (ME), in
this case z  s pˆ .
• The margin of error is tied to the level of
confidence. (Surveys in the newspaper or on
TV usually report a margin of error based on a
95% confidence level.)
• The more confident you want to be, the larger
the margin of error will be. More confidence
implies a wider interval.
• Since the standard error of pˆ is tied to the
sample size, as the sample size increases, the
value of the standard error deceases and
results in a narrower interval.
Things to Remember About a
Parameter
• A parameter does not vary – there is only
One true value of a parameter.
• You DO NOT KNOW the ture value of the
parameter. If you did, you would not need
a confidence interval.
• NEVER use pˆ when referring to the value
of the parameter, p.
Things to Remember About
Interpreting a Confidence Interval
• A confidence interval makes a statement about
the true population proportion, and your
interpretation of the interval should be about the
population proportion.
• Keep your interpretation to what you know – do
not claim to know too much. A sentence or two
should be sufficient.
• Sample statistics vary with each sample. Not
every interval you compute will capture the true
population proportion. Your interpretation of the
confidence interval should reflect how confident
you are that your interval has managed to
capture the true value; your confidence is in the
method used to produce the interval.
TESTING HYPOTHESES
ABOUT PROPORTIONS
Confidence Intervals and
Hypothesis Tests
• Confidence intervals, just one form of statistical
inference, allow us to use a sample statistic to
make a statement about how confident we are that
a population parameter lies within certain limits.
• Another form of statistical inference is a
hypothesis test or test of significance. We use
this form of statistical inference when a particular
conjecture or claim (hypothesis) has been made
about the true value of a population parameter.
• Again, we use sample statistics to help us decide
whether or not to believe the hypothesis. We want
to know if there is enough evidence to support
(NOT prove) the hypothesis or to reject it.
The Reasoning of Hypothesis
Testing
• Hypothesis
• The null hypothesis: The term null comes from the
idea that there is “no difference” between the
hypothesized value and the true population
parameter.
• H0: p=p0 (hypothesized value)
• The null hypothesis MUST have a sign of equality
and MUST be written using population parameters
and NEVER sample statistics.
• You MUST use standard notation (don’t make up
your own symbols).
• You should write your hypothesis in both symbols
and words.
• Your final decision MUST always be stated in terms
of rejecting or failing to reject the null hypothesis.
The Reasoning of Hypothesis
Testing (cont.)
• The alternative hypothesis: the
alternative hypothesis is what would be
accepted as true if the null hypothesis were
to be rejected. It can take one of three
different forms;
• For proportions, a one-sided alternative
would be either
Ha: p<p0 or Ha: p>p0.
• A two-sided alternative would be
Ha: p≠p0.
• The form used will depend on the study
being done and should be clear from
the context of the problem.
The Reasoning of Hypothesis
Testing (cont.)
• Plan
• Collect data that can be used to calculate a
sample statistic to test against the null
hypothesis.
• Decision Rule: Decide what will be
“convincing evidence” to reject the null
hypothesis.
• Mechanics
• Check the conditions for the sampling
distribution model.
• Calculate a test statistic. A test statistic is the
standardized value of the sample statistic
with respect to the hypothesized value.
The Reasoning of Hypothesis
Testing (cont.)
• In general, it has the form
Test statistic =
statistic - parameter
standard deviation of statistic
• For a proportion, the test statistic is given
by
 pˆ  p0  , where SD pˆ  p0 1  p0 
z
 
SD  pˆ 
n
• Note: When computing SD for the test
statistic, we use p0, because our model is
based on the conjecture or claim (null
hypothesis). When computing the SE for a
confidence interval, we use p-hat because
we do not have a hypothesized parameter.
The Reasoning of Hypothesis
Testing (cont.)
• Calculate the p-value.
• The p-value is the probability that a value
at least as extreme as the observed value
could occur if, in fact, the hypothesized
value were true.
• Use the normal table to convert your test
statistic to a p-value.
• The smaller the p-value, the less likely it is
that our sample came from a population
with the parameter value stated in our null
hypothesis. In other words, the smaller the
p-value, the stronger the evidence is
against the null hypothesis.
The Reasoning of Hypothesis
Testing (cont.)
• Conclusion
• Make a decision about whether to reject or fail
to reject the null hypothesis based upon the pvalue.
• Reject H0 if the calculated p-value is less than
the chosen α–level. We say that the result is
statistically significant.
• Your statement MUST include
• Decision (reject H0 in favor of Ha or fail to
reject H0).
• Criteria for decision (p-value compared with
α–level or test statistic compared with
critical value).
• Context (use the words of the problem to
restate your decision).
MORE ABOUT TESTS
Possible Errors When Hypothesis
Testing
• Because of the variability of sample data,
we may draw a wrong conclusion despite
having followed correct procedures.
• Whenever we make decisions based on
sample data, there is a risk of error.
• These errors are of two types, Type I and
Type II.
Type I Error
• A Type I Error is the consequence of
rejecting a null hypothesis that is in fact
true.
• The probability of making a Type I Error is
equal to the value of α, the level of
significance.
Type II Error
• A Type II Error is the consequence of
failing to reject a null hypothesis that is in
fact false.
• The probability of this type of error is
called β.
• The value of β changes depending on the
value of the parameter that is chosen as
the alternative.
• Note: Knowing how to calculate β is NOT a
requirement of the AP curriculum. You
ARE expected to understand the
difference between α and β and how they
affect each other.
Power
• The probability that a test will correctly
reject a null hypothesis that is in fact false
is called the power of the test.
• POWER = 1 - β.
• Power is the complement of a Type II
Error.
• High power is desired because it indicates
the sensitivity of the test to specific values
of the alternative hypothesis.
Ways to Reduce Error
• Type I Error (α) – decrease the α-level.
• Type II Error (β) – increase the α-level.
• Type I and Type II Errors – increase the
sample size, n.
• Power (1 - β) – increase the sample size,
n, or decrease β, or select a different,
more extreme alternative.
COMPARING TWO
PROPORTIONS
The Sampling Distribution for
 pˆ1  pˆ 2 
Conditions for the Difference of
Two Proportions
• Independence Condition – the two sample
groups are independent of one another.
• Randomization
• 10% Condition
• Success/Failure Condition
Two-Proportion z-Interval
Two Proportion z-Test
• The procedure for this test is essentially
the same as for a one-proportion z-test
with one slight difference: the null
hypothesis is that there is no difference
between the two proportions.
H0: p1=p2 (H0: p1-p2=0)
• The alternative hypothesis is determined
by the kind of difference you expect to get
in the particular situation (<, ≠, >).
Two Proportion z-Test
• The standard error in this case is pooled
(combine the two proportions). It is okay to
pool the proportions because the null
hypothesis assumes that p1=p2 and thus
the two populations are identical for the
attribute being studied.
SE pooled 
1 1
x x
pˆ pooled qˆ pooled    , where pˆ pooled  1 2
n1  n2
 n1 n2 
Two Proportion z-Test
It is highly unlikely that you will have to do this
calculation by hand. The calculator two-proportion
z-test calculates the z-score automatically and uses
the correct standard error.
What You Need to Know
• What is a sampling distribution.
• How to find means and standard
deviations for sampling distributions
(on formula sheet).
• How to calculate probabilities as it
relates to the CLT.
• What a confidence interval means
and how it is computed.
• What a confidence level means.
• What the margin of error means.
• How to compute a confidence
interval for proportions.
What You Need to Know
• How to use the ME to find a sample
size for proportions.
• How margin of errors change as the
confidence changes.
• What you can conclude from a CI
and what you can’t.
• The difference between the null and
alternate hypotheses.
• How to conduct a hypothesis test for
proportions.
• What a p-value means.
PRACTICE PROBLEMS
#1
Suppose that 35% of all business
executives are willing to switch companies
if offered a higher salary. If a headhunter
randomly contacts an SRS of 100
executives, what is the probability that
over 40% will be willing to switch
companies if offered a higher salary?
a)
b)
c)
d)
e)
.1469
.1977
.4207
.8023
.8531
#2
The average number of daily emergency
room admissions at a hospital is 85 with a
standard deviation fo 37. In an SRS of 30
days, what is the probability that the mean
number of daily emergency admissions is
between 75 and 95?
a)
b)
c)
d)
e)
.1388
.2128
.4090
.5910
.9474
#3
A confidence interval estimate is
determined from a SRS of n students.
Which of the following will result in a
smaller margin of error?
I. A smaller confidence level
II. A smaller sample size
a) I only
b) II only
c) both I and II
#4
A survey was conducted to determine the
percentage of high school students who planned to
go to college. The results were stated as 82% with
a margin of error of 5%. What is meant by +/- 5%?
a) Five percent of the population were not
surveyed.
b) In the sample, the percentage of students who
plan to go to college was between 77% and
87%
c) The percentage of the entire population of
students who plan to go to college is between
77% and 87%
d) It is unlikely that the given sample proportion
result would be obtained unless the true
percentage was between 77% and 87%
e) Between 77% and 87% of the population were
surveyed.
#5
A USA Today “Lifeline” column reported
that in a survey of 500 people, 39% said
they watch their bread while it’s being
toasted. Establish a 90% confidence
interval estimate for the percentage of
people who watch their bread being
toasted.
a) 39% +/- .078%
b) 39% +/- 2.2%
c) 39% +/- 2.8%
d) 39% +/- 3.6%
e) 39% +/- 4.3%
#6
A politician wants to know what
percentage of the voters support her
position on a hot issue. What size voter
sample should be obtained to determine
with 90% confidence the support level to
within 4%?
a)
b)
c)
d)
e)
21
25
423
600
1691
#7
Which of the following are true statements?
I. Hypothesis tests are designed to measure the
strength of the evidence against the null
hypothesis.
II. A well-planned test should result in a statement
either that the null hypothesis is true or that it is
false.
III. The alternate hypothesis is one-sided if there is
interest in deviations from the null hypothesis in
only one direction.
a)
b)
c)
d)
e)
I and II
I and III
II and III
I, II, and III
None of the above
#8
A building inspector believes that the percentage of
new construction with serious code violations may
be even greater than the previously claimed 7%.
She conducted a hypothesis test on 200 new
homes and finds 23 with serious code violations. Is
this strong evidence against the 7% claim?
a)
b)
c)
d)
e)
Yes, because the P-value is .0062
Yes, because the P-value is 2.5
No, because the P-value is only .0062
No, because the P-value is over 2
No, because the P-value is .045
#9
In a survey of 9700 T.V. viewers, 40% said
they watch network news programs. Find
the margin of error for this survey if we
want 95% confidence in our estimate of
the percent of T.V. viewers who watch
network news programs.
a) 1.12%
b) 1.28%
c) 0.731%
d) 0.975%
#10
Which is true about a 99% confidence interval
based on a given sample?
I. The interval contains 99% of the population.
II. Results from approximately 99% of all
samples will capture the true parameter in their
respective intervals.
III. The interval is wider than a 95% confidence
interval would be.
a) I only
b) II only
c) III only
d) II and III only
e) None
#11
To maintain customer satisfaction, an online catalog company wants to
have on-time delivery for 90% or better of the orders they ship. The
company has been shipping their orders via UPS and FedEx but will
switch to a new, cheaper delivery service called ShipFast unless there
is evidence that this service cannot meet the delivery goal of 90% or
better. As a test the company sends a random sample of orders via
ShipFast, and then makes follow-up phone calls to see if these orders
arrived on time. Which hypotheses should they test?
a) H0: p<0.90
Ha: p=0.90
b) H0: p>0.90
Ha: p=0.90
c) H0: p=0.90
Ha: p<0.90
d) H0: p=0.90
Ha: p≠0.90
e) H0: p=0.90
Ha: p>0.90
#12
A researcher investigating whether runners are
less likely to get colds than non-runners found a
P-value of 3%. This means that:
a) 3% of runners get colds.
b) 3% fewer runners get colds.
c) There’s a 3% chance that runners get fewer
colds.
d) There's a 3% chance our assumption of no
difference in number of colds whether a
runner or not is incorrect.
e) There’s a 3% chance that the sample statistic
or more extreme will occur assuming there is
no difference between number of colds
whether a runner or not.
Part Vi:
Chapters 23 - 25
LEARNING ABOUT THE
WORLD
INFERENCE ABOUT MEANS
The t-Distributions: A Sampling Distribution
for Means
• By the Central Limit Theorem (CLT), the means of repeated
samples tend to follow a normal model as long as the population
distribution is normal or the sample size is large enough.
• According to the CLT, the formula for the standard deviation of
the sampling distribution for means is

 x  SD( x ) 
n
• When we have to estimate with , the standard error of the
sampling distribution for means is
s
sx  SE ( x ) 
n
• Because σ is unknown, the standard error is used in place of the
standard deviation, and the shape of the sampling distribution
changes.
• The new model is a family of distributions called the t-distribution.
The t-Distributions
• The shape of the t-distributions (unimodal,
symmetric, and bell-shaped) is connected
to the sample size by a parameter called
the degrees of freedom.
• The increased variation due to small
sample size increases the probability in
the tails.
• The degrees of freedom (df) determine the
particular t-distribution.
• Degrees of freedom (df) equals (n – 1).
Sampling Distribution for Means
Conditions for Using
t-Distributions
• Randomization Condition
• 10% Condition
• Nearly Normal Condition – the data come from a
unimodal, symmetric, bell-shaped distribution. This can
be verified by constructing a histogram or a normal
probability plot of the sample data.
• If the population is approximately normal, the tstatistic is appropriate regardless of sample size.
• For n≤40, the t-statistic is appropriate provided
there are no outliers.
• For 15≤n<40, the t-statistic is appropriate provided
there are no outliers or strong skewness.
• For n<15, the t-statistic is appropriate provided the
distribution of the sample data is approximately
normal.
One-Sample Confidence Interval
for the Mean (σ unknown)
• Because we are using s to estimate , we
use a t-critical value for confidence
intervals and a t-test statistic for
hypothesis tests.
One-Sample t-Confidence
Interval
Hypothesis Test for Means
The format for all hypothesis tests is
essentially the same as it was for
proportions.
Determining Sample Size
• Once you have decided on a confidence level and
margin of error, you calculate the sample size from
the appropriate confidence interval formula.
• You should always round the sample size (n) UP to
the nearest integer. Sample size should be a whole
number
INFERENCE FOR TWO
INDEPENDENT SAMPLES
Comparing Two Means
• As is the case of proportions, the
parameter of interest is the difference
between two population means, μ1 - μ2.
The mechanics are the same.
The Sampling Distribution for Two
Independent Means  x1  x2 
• The mean of the sampling distribution of
 x1  x2  is μ1 - μ2.
• The standard error, because we
substitute s for σ, of the sampling
distribution is
SE  x1  x2  
 12
n1

 22
n2
• The assumptions and the conditions for
inference for two-sample means are the
same as those for one-sample means,
with the added condition that the samples
are independent.
A Two-Sample t-Interval for
Means
The formula for calculating the actual degrees of
freedom is rather complicated, and generally is
not a whole number (the TI calculator gives this
number).
A Two-Sample t-Interval for
Means
• When using the TI calculator, you are
asked to choose “Pooled: No Yes.”
Choose NO.
• The word pooled here does not have the
same meaning as it did for two
proportions.
Test for the Difference Between
Two Means
When you record a p-value, be certain that it is a
number from 0 to 1. Do not carelessly copy the
calculator output! Probabilities cannot exceed 1!
A SPECIAL CASE: TWO
SAMPLES OR ONE?
Matched Pairs
• What should we do when the two samples
are not independent.
• What if the data are for the same people at
two different times.
• It would not make sense to get averages
before and after, and do a two-sample ttest for means. Besides it would violate the
condition of independent samples.
• We really want to know the amount of
change for each individual. So we subtract
first and create a single set of data. Now
we use a one-sample procedure.
Matched Pairs
• There are many situations that call for data
to be treated in this way, but we must have
a good reason for pairing the data.
• Just because two samples have the same
number of data values is not a reason for
pairing them.
• It is up to you to recognize when a
matched pairs procedure is called for.
• For matched pairs, we use the same
procedures we used for the one-sample tinterval or the one-sample t-test for the
mean.
What You Need to Know
• Why the t-distribution is used to
model the sampling distribution of
sample means?
• Properties of the t-distribution, the
sampling distribution for means.
• What degrees of freedom is and
how to calculate.
• Conditions for using t-distribution.
• Calculate one-sample t-confidence
interval for means.
• Conduct one-sample t-hypothesis
test for means.
What You Need to Know
• Determine sample size for means.
• Sampling distribution for two means and
conditions required for inference.
• Calculate two-sample t-confidence interval
for means.
• Conduct two-sample t-hypothesis test for
the difference of means.
• Recognize a matched pairs situation.
• Calculate matched pairs t-confidence
interval and t-hypothesis test.
PRACTICE PROBLEMS
#1
In preparing to use a t procedure, suppose
we were not sure if the population was
normal. In which of the following
circumstances would we not be safe using a t
procedure?
a) A stemplot of the data is roughly bell
shaped.
b) A histogram of the data shows moderate
skewness.
c) A stemplot of the data has a large outlier.
d) The sample standard deviation is large.
#2
The weight of 9 men have mean of 175 lbs
and a standard deviation of 15 lbs. What is
the standard error of the mean?
a)
b)
c)
d)
e)
58.3
19.4
5
1.7
None of the above.
#3
What is the critical value t* which satisfies the
condition that the t distribution with 8 degrees
of freedom has probability 0.10 to the right of
t*?
a)
b)
c)
d)
e)
1.397
1.282
2.89
0.90
None of the above. The answer is _____.
#4
Suppose we have two SRSs from two distinct
populations and the sample are independent. We
measure the same variable for both samples.
Suppose both populations of the values of these
variables are normally distributed but the means
and the standard deviations are unknown. For
purposes of comparing the two means, we use
a)
b)
c)
d)
e)
Two-sample t procedures
Matched pairs t procedures
z procedures
The least-squares regression line
None of the above.
#5
• You want to compute a 90% confidence
interval for the mean of a population with
unknown population standard deviation.
The sample size is 30. The value of t* you
would use for this interval is
a)
b)
c)
d)
e)
f)
1.96
1.645
1.699
.90
1.311
None of the above.
#6
A 95% confidence interval for the mean
reading achievement score for a population
of third-graders is (44.2, 54.2). The margin of
error of this interval is
a)
b)
c)
d)
e)
95%
5
2.5
54.2
The answer cannot be determined from
the information given.
#7
Using the same set of data, you compute a
95% confidence interval and a 99%
confidence interval. Which of the following
statements is correct?
a)
b)
c)
d)
The intervals have the same width.
The 99% interval is wider.
The 95% interval is wider.
You cannot determine which interval is
wider until you know n and s.
#8
To use the two-sample t procedure to
perform a significance test on the difference
between two means, we assume
a) The populations’ standard deviations are
known.
b) The samples from each population are
independent.
c) The distributions are exactly normal in
each population.
d) The sample sizes are large.
e) All of the above.
#9
In a test for acid rain, 49 water samples
showed a mean pH level of 4.4 with a
standard deviation of 0.35. Find a 90%
confidence interval estimate for the mean pH
level.
a)
b)
c)
d)
e)
4.4±0.01175
4.4±0.0839
4.4±0.315
4.4±0.35
None of the above. The answer is _____.
Part VII:
Chapters 26 & 27
Inference When Variables Are Related
INFERENCE WHEN
VARIABLES ARE RELATED
• We have already reviewed inference for
proportions (z-intervals and tests) and
inference for means (t-intervals and tests).
• In this part of our review, we look at tests
for one and two categorical variables.
• The data for these tests are usually
presented in some form of table divided
into categories and must be in the form of
counts for each category.
COMPARING COUNTS
The Chi-Square Statistic
• There are three inference tests involving
categorical variables and counts.
• These tests introduce a new statistic called
the chi-squared statistic, which has the
same form for all three tests:
 
2

all cells
O  E 
2
E
• Where O stands for the observed value
and E stands for the expected value.
The Chi-Square Distribution
• The χ2 distribution is a family of distributions.
• Each individual distribution is specified by its
degrees of freedom.
• Unlike the t-distribution, the value of n in this
case is not the sample size but rather the
number of categories.
• The χ2 distribution is skewed to the right and
has only positive values.
• Because there are only positive values, all
hypothesis tests are right tailed tests and the
p-value is computed by calculating the area
under the curve to the right of the computed
χ2 test statistic.
THE CHI-SQUARE TESTS
Goodness-of-Fit Test
• One categorical variable – one sample.
• Used to determine how well a set of observed values
matches a set of expected values.
• In other words, how well does a sample distribution
match the hypothesized population distribution?
• The degree of freedom is the number of categories
minus 1 (#cat. – 1).
• The expected values (E=np) are calculated in one of
two ways:
• Values for each category are expected to all have
a uniform distribution.
• Values are expected to follow a distribution
specified according to some stated condition.
Assumptions/Conditions for Chi-Square
Test for Goodness-of-Fit
• Counted Data Condition: Make sure the
data are listed in the form of counts for
each category. Precents or proportions
WON’T DO – convert them to counts
(rounded to whole numbers) if necessary.
• Randomization Condition: The individual
cases should be a random sample from
the population of interest.
• Expected Cell-Frequency Condition:
There are at least 5 cases in each
expected cell.
Chi-Square Test for Goodness-of-Fit
• Null and alternative hypotheses are
usually written in words.
• The null hypothesis is a statement that the
distributions are consistent or the same.
• The alternative hypothesis is a statemnt
that the distributions are not consistent or
the same.
Chi-Square Test for Goodness-of-Fit
Chi-Square Test for Goodness-of-Fit
• If the expected distribution is uniform, the
percentage would be 100%/# of cells. For
example, we expect digits in a randomnumber table to appear with equal
frequencies. The percentage for each digit
would be 10%. We would use these
percentages to calculate the expected
counts.
• If you are asked to identify the cell that
contributes the most to the χ2 test statistic,
just look at the individual components that
contribute to your sum.
Test of Independence
• The “Chi-Square 2 Test of Independence”
is used to determine if, in a single
population, there is an association
between two categorical variables.
• The data are presented in a two-way table.
• The null and alternative hypotheses are
written in words.
• The null hypothesis is a statement that the
two categories are independent or not
related.
• The alternative hypothesis is that the two
categories are depentent or related.
Chi-Square Test of Independence
• The Expected Values are calculated as;
E
 row total  column total 
table total
• The degrees of freedom are calculated as;
df   # rows  1 # columns  1
• Although, we reject the null hypothesis of
independence, we cannot interpret our
small p-value as proof of causation. A
failure of independence does not prove a
cause-and-effect relationship. There may
be lurking variables present.
What You Need to Know
• Be able to write hypotheses for all of the
various χ2 tests.
• Be able to do the various tests with and
without your calculator, decide to
reject/retain, and write a conclusion.
• Know how to find expected values for chisquare tests.
PRACTICE PROBLEMS
#1
Answer: B
#2
A test of independence for data organized in a
two-way table relating number of siblings and
number of family relocations is conducted
using the chi-square distribution. The p-value
of the test is .045. If alpha is .05, then which of
the following is a valid conclusion of the test?
a) The mean is significant.
b) We reject the hypothesis that the variables
are dependent.
c) We accept the hypothesis that the variables
are independent.
d) We have sufficient evidence to reject the
hypothesis that the variables are independent.
e) The variables are independent.
#3
A chi-square goodness of fit test is used to
test whether a 0-9 spinner is “fair” (ie., the
outcomes are all equally likely). The spinner
is spun 100 times, and the results are
recorded. Which member of the chi-square
family of curves is used?
a)
b)
c)
d)
e)
χ2(8)
χ2(9)
χ2(10)
χ2(99)
None of the above
#4
A two-way table of counts is analyzed to
examine the hypothesis that the row and
column classifications are independent.
There are 3 rows and 4 columns. The
degrees of freedom for the chi-square
statistic are
a)
b)
c)
d)
e)
12
11
6
The minimum of n1-1 and n2-1
None of the above
#5
In the paper “Color
Association of Male
and Female 4th grade
School Children”,
children were asked
to indicate what
emotion they
associated with the
color red.
The expected frequency for the cell
corresponding to Anger and Males is:
a)
b)
c)
d)
e)
15.9
55.7
30.4
31.9
29.1
#6
In the paper “Color
Association of Male
and Female 4th grade
School Children”,
children were asked
to indicate what
emotion they
associated with the
color red.
The null hypothesis is:
a) Emotional association with red is independent of gender.
b) Gender is dependent upon the emotional association with
red.
c) The probability of selecting an emotion with red is related
to gender.
d) The number of children in each cell does not depend upon
gender or upon emotion.
e) The color red is independent of the emotion associated
with it and with gender.
#7
In the paper “Color
Association of Male
and Female 4th grade
School Children”,
children were asked
to indicate what
emotion they
associated with the
color red.
The null hypothesis will be rejected at α=0.05 if
the test statistic exceeds:
a)
b)
c)
d)
e)
3.84
5.99
7.81
9.49
14.07
#8
In the paper “Color
Association of Male
and Female 4th grade
School Children”,
children were asked
to indicate what
emotion they
associated with the
color red.
The approximate P-value is:
a) Between 0.100 and 0.900
b) Between 0.050 and 0.100
c) Between 0.025 and 0.050
d) Between 0.010 and 0.025
e) Between 0.005 and 0.010