Download 7.2 Day 2

Section 7.2
Sampling Distribution of
the Sum of a Sample
Properties of Sampling Distribution
of the Sample Mean
If a random sample of size n is selected
from a population with mean  and
standard deviation  , then what are the
three basic properties of the sampling
distribution of the sample mean for center,
spread, and shape?
Properties of Sampling Distribution
of the Sample Mean
Center
x = 
Properties of Sampling Distribution
of the Sample Mean
Spread


X =
n
Spread decreases as sample size increases
Properties of Sampling Distribution
of the Sample Mean
Shape
The shape of the sampling distribution will
be approximately normal if the population
is approximately normal.
For other populations, the sampling
distribution becomes more normal as n
increases (Central Limit Theorem).
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of
20 families?
µ = 0.9
σ = 1.1
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of 20
families?
Reasonably likely values are those in the
middle 95% of the sampling
distribution.
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of 20
families?
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of 20
families? µ = 0.9, σ = 1.1
For approx. normal distributions,
reasonably likely outcomes are those
within 1.96 standard errors of the mean.
 x  1.96( SE )
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of
20 families?
0.9  1.96(0.25)
which gives an average number of children
between 0.41 and 1.39
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The shape of the sampling distribution will
be approximately normal because the
population’s distribution is approximately
normal.
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The shape of the sampling distribution will
be approximately normal because the
population’s distribution is approximately
normal.
The mean of the sampling distribution will be
equal to the population mean of .266.
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The shape of the sampling distribution will
be approximately normal because the
population’s distribution is approximately
normal.
The mean of the sampling distribution will be
equal to the population mean of .266.
The standard error is:
Think
Problems we’ve worked so far involved:
• Shape, center, and spread of sampling
distributions of means of various sample
sizes
-- Use the three properties we discussed
Think
Problems we’ve worked so far involved:
• Probability a random sample of n families
will have an average of x children or less
normalcdf(lower bound, upper bound,
mean of sampling distribution, standard
error)
Think
Problems we’ve worked so far involved:
• What average numbers of ??? are
reasonably likely in a random sample of
n things?
μx ± 1.96(SE)
Think
What if we have problems involving sample
totals such as if you pick 15 households
at random, what is the probability that they
have at least 30 motor vehicles among
them?
Properties of the Sampling Distribution
of the Sum of a Sample
Three properties based on the following
premise: If a random sample of size n is
selected from a distribution with mean 
and standard deviation  , then:
Properties of the Sampling
Distribution of the Sum of a Sample
Three properties based on the following
premise: If a random sample of size n is
selected from a distribution with mean 
and standard deviation  , then:
(1) The mean of the sampling distribution of
the sum is  sum = n 
Properties of the Sampling
Distribution of the Sum of a Sample
Three properties based on the following
premise: If a random sample of size n is
selected from a distribution with mean 
and standard deviation  , then:
(2) The standard error of the sampling
distribution of the sum is
 sum = n  
Properties of the Sampling
Distribution of the Sum of a Sample
Three properties based on the following premise:
If a random sample of size n is selected from a
distribution with mean  and standard
deviation  , then:
(3) The shape of the sampling distribution
will be approximately normal if the
population is approximately normally
distributed. For other populations the
sampling distribution will become more
normal as n increases.
The distribution of the number of motor
vehicles per household in the U.S. is
roughly symmetric with mean 1.7 and
standard deviation 1.0.
The distribution of the number of motor
vehicles per household in the U.S. is
roughly symmetric with mean 1.7 and
standard deviation 1.0.
(a) If you pick 15 households at random,
what is the probability that they have at
least 30 motor vehicles among them?
The distribution of the number of motor
vehicles per household in the U.S. is
roughly symmetric with mean 1.7 and
standard deviation 1.0.
(a) If you pick 15 households at random,
what is the probability that they have at
least 30 motor vehicles among them?
Use normalcdf (lower bound, upper bound,
mean of the sum, standard deviation of
sum)
The distribution of the number of motor
vehicles per household in the U.S. is
roughly symmetric with mean 1.7 and
standard deviation 1.0.
(a) If you pick 15 households at random,
what is the probability that they have at
least 30 motor vehicles among them?
normalcdf (30, 1E99, nμ, n ●σ )
The distribution of the number of motor
vehicles per household in the U.S. is
roughly symmetric with mean 1.7 and
standard deviation 1.0.
(a) If you pick 15 households at random,
what is the probability that they have at
least 30 motor vehicles among them?
normalcdf (30, 1E99, (15 ● 1.7), ( 15 1 ) )
≈ 0.1226
The distribution of the number of motor
vehicles per household in the U.S. is
roughly symmetric with mean 1.7 and
standard deviation 1.0.
(b) If you pick 20 households at random,
what is the probability that they have
between 25 and 30 motor vehicles among
them?
The distribution of the number of motor vehicles
per household in the U.S. is roughly symmetric
with mean 1.7 and standard deviation 1.0.
(b) If you pick 20 households at random,
what is the probability that they have
between 25 and 30 motor vehicles among
them?
normalcdf (25, 30, (20 ● 1.7),( 20 1 ) )
≈ 0.1635
Suppose a population distribution of the
number of children per family in the U.S.
has a mean of 0.9 and a standard
deviation of 1.1.
(a) Do you think it is reasonably likely that a
sample of 1000 households will produce at
least 1000 children?
Suppose a population distribution of the
number of children per family in the U.S.
has a mean of 0.9 and a standard
deviation of 1.1.
(a) Do you think it is reasonably likely that a
sample of 1000 households will produce
at least 1000 children?
normalcdf (1000, 1E99, (1000●0.9),
10001.1) ≈ 0.002; so not reasonably likely
Suppose a population distribution of the
number of children per family in the U.S.
has a mean of 0.9 and a standard
deviation of 1.1.
(a) Do you think it is reasonably likely that a
sample of 1000 households will produce
at least 1000 children?
normalcdf (1000, 1E99, (1000●0.9),
10001.1) ≈ 0.002; so not reasonably likely
Need P(> 1000 children) to be > 0.025
Suppose a population distribution of the
number of children per family in the U.S.
has a mean of 0.9 and a standard
deviation of 1.1.
(b) Suppose now we have a random sample
of 1200 households. Does this
dramatically improve the chances of
seeing at least 1000 children in the
sampled households?
Suppose a population distribution of the
number of children per family in the U.S.
has a mean of 0.9 and a standard
deviation of 1.1.
(b) Suppose now we have a random sample
of 1200 households. Does this
dramatically improve the chances of
seeing at least 1000 children in the
sampled households? Yes
normalcdf(1000, 1E99, (1200 ● 0.9),
12001.1 ) ≈ 0.982
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The population standard deviation is 2.402.
Now do part b.
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d. The rule that about 95% of the
observations lie within approximately two
standard errors of the population mean
works well for n = 25 as this distribution is
approximately normal.
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normalcdf(510, 1E99, 500, 50) ≈ .4207
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normalcdf(510, 1E99, 500, 20) ≈ .3085
Questions?