Sampling Distributions

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Chapter 9

• Sampling Distributions

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Parameter

• A number that describes the population
• Symbols we will use for parameters include
• m - mean
• s standard deviation
• p or P proportion (p)
• a y-intercept of LSRL
• b slope of LSRL

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Statistic

• A number that can be computed from sample data
  that can estimate a parameter
• Symbols we will use for statistics include
• x mean
• s standard deviation
• p proportion
• a y-intercept of LSRL
• b slope of LSRL

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Identify the boldface values as parameter or
statistic.
A carload lot of ball bearings has mean diameter
2.5003 cm. This is within the specifications for
acceptance of the lot by the purchaser. By
chance, an inspector chooses 100 bearings from
the lot that have mean diameter 2.5009 cm.
Because this is outside the specified limits, the
lot is mistakenly rejected.
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Why do we take samples instead of taking a census?

• A census is not always accurate.
• Census are difficult or impossible to do.
• Census are very expensive to do.

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A distribution is all the values that a variable
can be.
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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.
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 Consider the population the length of fish (in
inches) in my pond - consisting of the values 2,
7, 10, 11, 14
What is the mean and standard deviation of this
population?
mx 8.8
sx 4.0694
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Lets take samples of size 2 (n 2) from
this population How many samples of size 2 are
possible?
5C2 10
mx 8.8
Find all 10 of these samples and record the
sample means.
What is the mean and standard deviation of the
sample means?
sx 2.4920
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Repeat this procedure with sample size n 3 How
many samples of size 3 are possible?
5C3 10
mx 8.8
What is the mean and standard deviation of the
sample means?
Find all of these samples and record the sample
means.
sx 1.6613
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What do you notice?

• The mean of the sampling distribution EQUALS the
  mean of the population.
• As the sample size increases, the standard
  deviation of the sampling distribution decreases.

as n
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A statistic used to estimate a parameter is
unbiased if the mean of its sampling distribution
is equal to the true value of the parameter being
estimated.
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Internet Activity drawing samples
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General Properties

• Rule 1
• Rule 2
• This rule is approximately correct as long as no
  more than 10 of the population is included in
  the sample

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General Properties

• Rule 3
• When the population distribution is normal, the
  sampling distribution of x is also normal for
  any sample size n.

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General Properties

• Rule 4 Central Limit Theorem
• When n is sufficiently large, the sampling
  distribution of x is well approximated by a
  normal curve, even when the population
  distribution is not itself normal.

CLT can safely be applied if n exceeds 30.
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Normal DistributionsReview

• Symmetrical bell-shaped (unimodal) density curve
• Above the horizontal axis
• N(m, s)
• The transition points occur at m s
• Probability is calculated by finding the area
  under the curve
• As s increases, the curve flattens
  spreads out
• As s decreases, the curve gets
  taller and thinner

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Normal distributions occur frequently.

• Length of newborn child
• Height
• Weight
• ACT or SAT scores
• Intelligence
• Number of typing errors
• Chemical processes

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A
B
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Do these two normal curves have the same mean?
If so, what is it? Which normal curve has a
standard deviation of 3? Which normal curve has a
standard deviation of 1?
YES
B
A
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Empirical Rule

• Approximately 68 of the observations fall within
  s of m
• Approximately 95 of the observations fall within
  2s of m
• Approximately 99.7 of the observations fall
  within 3s of m

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Remember the army helmets . . .

• 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 soldiers head will have a circumference
  that is greater than 23.5 inches?

P(X gt 23.5) .2623
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• 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 to part (a)? Explain
What normal curve are you now working with?
.0774
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Suppose a team of biologists has been studying
the Pinedale childrens 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 lt X lt 12) .8427
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What is the probability that the mean length of
five trout taken at random is between 8 and 12
inches long? What sample mean would be at the
95th percentile? (Assume n 5)
Do you expect the probability to be more or less
than the answer to part (a)? Explain
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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?
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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?
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Does this result indicate that the manufacturers
claim is incorrect?
Yes, not likely to happen by chance alone.