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Chapter 12
Section 1
Sampling Distributions
Parameter
• A number that describes the population
• Symbols we will use for parameters
include
m - mean
s – standard deviation
p – proportion (p)
a – y-intercept of LSRL
b – slope of LSRL
Statistic
• A number that can be computed from
sample data without making use of any
unknown parameter
• Symbols we will use for statistics include
x – mean
s – standard deviation
p – proportion
a – y-intercept of LSRL
b – slope of LSRL
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. It is
typically represented as a
probability distribution in the
form of a table, a histogram or a
formula.
Consider the population of 5
fish in my pond – the length of
fish (in inches):
2, 7, 10, 11, 14
What is the mean
mand
8.8
x =standard
deviation
of this
sx = 4.0694
population?
Let’s take samples of size 2
(n = 2) from this population:
How many samples of size 2 are
possible?
C = 10
5
2
mx = 8.8
sx = 2.4919
Find
all 10
of
What
is the
mean
these
and samples
standardand
record
theofsample
deviation
the
means.
sample
means?
Repeat this procedure with sample
size n = 3
How many samples of size 3 are
possible?
C = 10
5
mx = 8.8
sx =
3
What
mean
Find
allisofthe
these
and standard
samples
and
deviation
of the
record
the
sample
1.66132 sample means?
means.
What do you notice?
• The mean of the sampling distribution
EQUALS the mean of the population.
mx = m
• As the sample size increases, the standard
deviation of the sampling distribution
decreases.
as n
sx
A statistic used to estimate a
parameter is unbiased if the
mean of its sampling
distribution
is
equal
to
the
Remember the Jelly Blubbers?
true value
of the parameter
The judgmental
samples were
centered
too high & were bias, while
being
estimated.
the randomly selected samples were
centered over the true mean
General Properties
Rule 1:
mx = m
s
Rule 2: sx =
n
This rule is approximately correct as long
as no more than 10% of the population is
included in the sample
General Properties
Rule 3:
When the population distribution is
normal, the sampling distribution
of x is also normal for any sample
size n.
General Properties
Rule 4: Central Limit Theorem (CLT)
When n is sufficiently large, the
sampling distribution of x is well
approximated by a normal curve, even
when the population distribution is not
How large is “sufficiently large”
itself normal.
anyway?
CLT can safely be applied if n exceeds 30.
Central Limit Theorem
1. If the samples of size n, n ≥ 30 are
drawn from any population with a
mean µ and a standard deviation σ
then the sampling distribution of the
sample means approximates a normal
distribution. The greater the sample
size, the better the approximation.
Central Limit Theorem
2. If the population is normally
distributed, the sampling
distribution of sample means is
normally distributed for any
sample of size n.
Central Limit Theorem
To apply the central limit theorem we
need to represent each value by a zscore:
z
x  mx
sx

x  mx
s
n
Central Limit Theorem
Note: the notation also changes
Before we had P(x > 35)
Now we have P( x  35)
EX) The army reports that the distribution of
head circumference among soldiers is
approximately normal with mean 22.8 inches and
standard deviation of 1.1 inches.
a) What is the probability that a randomly
selected soldier’s head will have a circumference
that is greater than 23.5 inches?
P(X > 23.5) = .2623
b) What is the probability that a random
sample of five soldiers will have an
average head circumference that is greater
than 23.5 inches?
Do you expect the probability to
be more or
less
than
the
answer
What normal curve are
to part
(a)?
Explain
you now working with?
P(X > 23.5) = .0774
Suppose a team of biologists has been
studying the Pinedale children’s fishing
pond. Let x represent the length of a single
trout taken at random from the pond. This
group of biologists has determined that the
length has a normal distribution with mean of
10.2 inches and standard deviation of 1.4
inches. What is the probability that a single
trout taken at random from the pond is
between 8 and 12 inches long?
P(8 < X < 12) = .8427
What is the probability that the mean
length of five trout taken at random is
between 8 and 12 inches long?
Do xyou
expect
the probability to
P(8<
<12)
= .9978
be more or less than the answer
to part (a)? Explain
What sample mean would be at the 95th
percentile? (Assume n = 5)
x = 11.23 inches
A soft-drink bottler claims that, on average,
cans contain 12 oz of soda. Let x denote the
actual volume of soda in a randomly selected
can. Suppose that x is normally distributed
with s = .16 oz. Sixteen cans are randomly
selected and a mean of 12.1 oz is calculated.
What is the probability that the mean of 16
cans will exceed 12.1 oz?
P(x >12.1) = .0062
A hot dog manufacturer asserts that one of its
brands of hot dogs has a average fat content of
18 grams per hot dog with standard deviation of
1 gram. Consumers of this brand would
probably not be disturbed if the mean was less
than 18 grams, but would be unhappy if it
exceeded 18 grams. An independent testing
organization is asked to analyze a random
sample of 36 hot dogs. Suppose the resulting
sample mean is 18.4 grams. What is the
probability that the sample mean is greater
than 18.4 grams?
P(x >18.4) = .0082
Does this result indicate that the
manufacturer’s claim is incorrect?
Yes, not likely to happen by
chance alone.