Transcript Sample Means - Walton High

Section 9.3
Sample Means
Sample Means
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The mean and standard deviation of a
population are parameters  we write these
as µ and σ
The mean and standard deviation calculated
from sample data are statistics  we write
these as x and s
Sampling Distribution of a sample
mean
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Draw an SRS of size n from a population that
has the normal distribution with mean µ and
standard deviation σ. Then the sample

mean x has the normal distribution N(µ,
)
n
THE CENTRAL LIMIT THEOREM
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Central Limit Theorem states that for large n
the sampling distribution of x is approximately
normal for any population with finite standard
deviation σ. The mean and standard deviation of
the normal distribution are the mean µ and

standard deviation .
n
Like before (with proportions), we want our
population to be at least ten times as big as the
sample to use this recipe for standard deviation
Law of Large Numbers
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Law of Large Numbers states that the
actually observed mean outcome x of a
large number of observations must
approach the mean µ of the population as
the number of observations increases.
Let’s Try Some Multiple Choice
NUMBER ONE
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When a sample is randomly selected from a
population, what is most likely to occur?
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A. The sample distribution will be identical to the
population distribution.
B. The sample distribution will be quite similar to the
population distribution at first, but will become less similar
as the sample gets larger.
C. The sample distribution could be quite different from the
population distribution at first, but will become more similar
as the sample gets larger.
D. The sample distribution will not be at all similar to the
population distribution.
AND THE ANSWER IS. . .
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Correct Answer: C — The sample distribution could
be quite different from the population distribution at
first, but will become more similar as the sample gets
larger.
Explanation: Small samples are often quite different
from the populations they are from, but as the
sample becomes larger the sample tends to
resemble the population distribution more and more.
This is true in terms of the overall shape of the
distribution, as well as the sample mean compared
with the population mean.
Let’s Try Some Multiple Choice
NUMBER TWO
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When samples are randomly selected from a
population, what is most likely to occur?
–
–
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A. The sampling distribution will be identical to the
population distribution.
B. The sampling distribution will be quite similar to the
population distribution at first, but will become more
normal as the samples gets larger.
C. The sampling distribution could be quite different from
the population distribution at first, but will become more
similar as the samples gets larger.
D. The sampling distribution will not be at all similar to the
population distribution.
AND THE ANSWER IS. . .
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Correct Answer: B — The sampling distribution will
be quite similar to the population distribution at first,
but will become more normal as the samples get
larger.
Explanation: Don’t forget, the sampling distribution is
different than a sample distribution. The Central
Limit Theorem reminds us that the sampling
distribution will be approximately normal for large
sample sizes regardless of the shape of the
population distribution. This is what makes the
Central Limit Theorem so important!
NUMBER THREE
Let’s Try Some Multiple Choice
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Suppose you randomly selected a group of American
adults and asked them how many hours of TV they
watched this past
Saturday, and the
average of their
answers was 4
hours. Which of the
following would be
most likely to be
the distribution of
their answers?
AND THE ANSWER IS. . .
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Correct Answer: D — Distribution D
Explanation: Distribution D shows most responses
between 0 and 5 hours, with a short tail on the right.
Distribution A is not likely because it is extremely
unlikely that the number of people who watch 8
hours of TV in a day is the same as the number of
people who watch 1 hour of TV. Distribution B is too
tightly bunched; surely not everyone you sample
would have watched 3, 4, or 5 hours or TV. Finally,
distribution C is impossible because it appears that
someone in the sample responded that he or she
watched 25 hours of TV on Saturday.
NUMBER FOUR
Let’s Try Some Multiple Choice
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Which of the sample distributions on the next
slide is most likely to be from this population
distribution?
Choices for Number Four
AND THE ANSWER IS. . .
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Correct Answer: D — Sample D
Explanation: Only sample D resembles the given
population sample. Even though sample A has
exactly the same mean as the population, the values
in sample A are too spread out (ranging from 0 to
20). Sample B is has only the values 9, 10, and 11 in
it, and that is highly unlikely to come from the given
population. Sample C appears to have far more
values near 5 and 15, and fewer near 10, than the
population.
NUMBER FIVE
Let’s Try Some Multiple Choice
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Which of the following population
distributions is more likely to be well
represented by a very small sample?
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A. The top (red) distribution would be
better represented by a small sample
because it is more spread out.
B. The bottom (blue) distribution would
be better represented by a small sample
because it is less spread out.
C. There should be no difference in how
well the two populations are
approximated by a small sample as long
as the samples are randomly selected.
D. The two cannot be compared
because their means are different.
AND THE ANSWER IS. . .
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Correct Answer: B — The bottom (blue) distribution
would be better represented by a small sample
because it is less spread out.
Explanation: The bottom (blue) distribution is fairly
tightly clustered, so even a small sample should
produce a mean that is fairly near the true population
mean. The top (red) distribution is much more
spread out, so there is a greater chance that a small
sample could select values far from the center of the
distribution. This would cause the sample mean to
be relatively far from the population mean.
(pp. 494- 495 # 9.38)
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And One Free Response
Judy’s doctor is concerned that she may suffer from
hypokalemia (low potassium in the blood). There is
variation both in the actual potassium level and in the blood
test that measures the level. Judy’s measured potassium
level varies according to the normal distribution with μ=3.8
and σ =0.2. A patient is classified as hypokalemic if the
potassium level is below 3.5.
a) If a single potassium measurement is made, what is the
probability that Judy is diagnosed as hypokalemic?
normalcdf(-E99, 3.5, 3.8, 0.2) ≈ 0.0668
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b) If measurements are made instead on 4 separate days
and the mean result is compared with the criterion 3.5, what
is the probability Judy is diagnosed as hypokalemic?
normalcdf(-E99, 3.5, 3.8, 0.2/√(4)) ≈ 0.0013