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Check your Skills – Chapter 12
 Calculate the probabilities for the
following using this table:
1.
A = the victim was male
Accidents
Homicide
Suicide
Female
1818
457
345
2620
Male
6457
2870
2152
11479
Total
8275
3327
2497
14099
P(A) = 11479/14099 = 0.81
2.
A = the victim was male, given B = the death was accidental
P(A|B) = P(A and B)/P(B) = (6457/14099)/(8275/14099) = 6457/8275 = .78
3.
That the death was accidental, given the victim was male
P(B|A) = P(B and A)/P(A) = (6457/14099)/(11479/14099) = 6457/11479 = .56
4.
Let A be the event that a victim of violent death was a woman and B the
event that the death was a suicide. Find the proportion of suicides among
violent deaths of women
P(B|A) = P(B and A)/P(A) = (345/14099)/(2620/14099) = 345/2620 = .13
CHAPTER 12
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EXERCISE 12.31
 Getting into college. Ramon has applied to both Princeton and
Stanford. He thinks the probability that Princeton will admit him is 0.4,
the probability that Stanford will admit him is 0.5, and the probability
that both will admit him is 0.2.
 Make a Venn diagram. Then answer these questions.
 (a) What is the probability that neither university admits Ramon?
 (b) What is the probability that he gets into Stanford but not Princeton?
 (c) Are admission to Princeton and admission to Stanford independent
events?
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CHAPTER 11
Sampling Distributions
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SAMPLING TERMINOLOGY
Parameter
 fixed, unknown number that describes the population
Statistic
 known value calculated from a sample
 a statistic is often used to estimate a parameter
Variability
 different samples from the same population may yield
different values of the sample statistic
Sampling Distribution
 tells what values a statistic takes and how often it takes
those values in repeated sampling
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PARAMETER VS. STATISTIC
A properly chosen sample of 1600 people across
the United States was asked if they regularly
watch a certain television program, and 24% said
yes. The parameter of interest here is the true
proportion of all people in the U.S. who watch the
program, while the statistic is the value 24%
obtained from the sample of 1600 people.
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PARAMETER VS. STATISTIC
mean of a population is denoted by µ – this is a
parameter.
 The
mean of a sample is denoted by x – this is a
statistic. x is used to estimate µ.
 The
 The
true proportion of a population with a certain trait is
denoted by p – this is a parameter.
 The
proportion of a sample with a certain trait is
denoted by p̂ (“p-hat”) – this is a statistic. p̂ is used to
estimate p.
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THE LAW OF LARGE NUMBERS
Consider sampling at random from a population
with true mean µ. As the number of
(independent) observations sampled, n increases,
the mean of the sample gets closer and closer to
the true mean of the population.
As n gets larger and larger, x approaches the value of
µ
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THE LAW OF LARGE NUMBERS GAMBLING
 The “house” in a gambling operation is not gambling at all
 the games are defined so that the gambler has a negative
expected gain per play (the true mean gain after all
possible plays is negative)
 each play is independent of previous plays, so the law of
large numbers guarantees that the average winnings of a
large number of customers will be close the the (negative)
true average
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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 (n) from the same population
 to describe a distribution we need to specify the shape,
center, and spread
 we will discuss the distribution of the sample mean (x-bar)
in this chapter
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CASE STUDY
Does This Wine Smell Bad?
Dimethyl sulfide (DMS) is sometimes present in
wine, causing “off-odors”. Winemakers want to
know the odor threshold – the lowest concentration
of DMS that the human nose can detect. Different
people have different thresholds, and of interest is
the mean threshold in the population of all adults.
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CASE STUDY
Does This Wine Smell Bad?
Suppose the mean threshold of all adults is =25
micrograms of DMS per liter of wine, with a
standard deviation of =7 micrograms per liter
and the threshold values follow a bell-shaped
(normal) curve.
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WHERE SHOULD 95% OF ALL INDIVIDUAL
THRESHOLD VALUES FALL?
mean plus or minus two standard deviations
25  2(7) = 11
25 + 2(7) = 39
95% should fall between 11 & 39
What about the mean (average) of a sample of n
adults? What values would be expected?
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SAMPLING DISTRIBUTION
What about the mean (average) of a sample of n
adults? What values would be expected?

Answer this by thinking: “What would happen if we took many samples
of n subjects from this population?” (let’s say that n=10 subjects make
up a sample)
– take a large number of samples of n=10 subjects from the
population
– calculate the sample mean (x-bar) for each sample
– make a histogram (or stemplot) of the values of x-bar
– examine the graphical display for shape, center, spread
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CASE STUDY
Does This Wine Smell Bad?
Mean threshold of all adults is =25 micrograms per liter,
with a standard deviation of =7 micrograms per liter and
the threshold values follow a bell-shaped (normal) curve.
Many (1000) repetitions of sampling n=10 adults from the population
were simulated and the resulting histogram of the 1000
x-bar values is on the next slide.
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CASE STUDY
Does This Wine Smell Bad?
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MEAN AND STANDARD DEVIATION
OF SAMPLE MEANS
If numerous samples of size n are taken from
a population with mean  and standard
deviation  , then the mean of the sampling
distribution of X is  (the population mean)
and the standard deviation is: 
n
( is the population s.d.)
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MEAN AND STANDARD DEVIATION
OF SAMPLE MEANS
the mean of X is , we say that X is an unbiased
estimator of 
 Since
observations have standard deviation ,
but sample means X from samples of size n have
standard deviation 
.
Individual
n
Averages are less variable than individual observations
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SAMPLING DISTRIBUTION OF
SAMPLE MEANS
If individual observations have the N(µ, )
distribution, then the sample mean X of n
independent observations has the N(µ, / n )
distribution.
“If measurements in the population follow a Normal
distribution, then so does the sample mean.”
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CASE STUDY
Does This Wine Smell Bad?
Mean threshold of
all adults is =25
with a standard
deviation of =7,
and the threshold
values follow a
bell-shaped
(normal) curve.
CHAPTER 11
(Population distribution)
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CENTRAL LIMIT THEOREM
If a random sample of size n is selected from ANY
population with mean  and standard deviation  ,
then when n is large the sampling distribution of
the sample mean X is approximately Normal:
X is approximately N(µ, / n )
“No matter what distribution the population values follow, the
sample mean will follow a Normal distribution if the sample size is
large.”
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CENTRAL LIMIT THEOREM:
SAMPLE SIZE

How large must n be for the CLT to hold?
– depends on how far the population distribution is from
Normal
 the further from Normal, the larger the sample size
needed
 a sample size of 25 or 30 is typically large enough for
any population distribution encountered in practice
 recall: if the population is Normal, any sample size will
work (n≥1)
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CENTRAL LIMIT THEOREM:
SAMPLE SIZE AND DISTRIBUTION OF X-BAR
CHAPTER 11
n=1
n=2
n=10
n=25
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