Sampling Distributions

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• Chapter 7
• Sampling Distributions

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Introduction

• For the normal distribution, the location and
  shape are described by µ and s.
• For a binomial distribution consisting of n
  trials, the distribution is determined by p.
• These parameters specify the exact form of a
  distribution.
• Often the values of parameters are unknown.

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Example

• We believe that the height of UNI students is
  approximately normally distributed, but µ and s
  are unknown.

We need to rely on the sample to learn about
these parameters.
We may consider using the sample mean as
a possible estimator of parameter µ, and is
called a statistic for µ.
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Sampling Distributions

• Parameter calculated from the sample are called
  statistics.
• Statistics take values varying from sample to
  sample, and hence are random variables, e.g.,
• sample mean is a random variable.
• The probability distributions of statistics,
  e.g., , are called sampling distributions.

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Example
Population 3, 5, 2, 1. Draw samples of size n
3 without replacement. What is the probability
distributions of sample mean statistics
?
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Example
Each value is equally likely, with probability 1/4
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Example
Toss a fair die once (n 1). The distribution of
x, the number on the upper face is
Mean
3.5
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Example
.

• flat or uniform.

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Example
Toss a fair die twice (n 2). What is the
distribution of , the average number on the
two upper faces?
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The possible values of
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(No Transcript)
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Example
Toss a fair coin n 3 times. The distribution of
, the average number on the two upper faces
is approximately normal.
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Sampling distribution of
sample size n2
sample size n1
sample size n3
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Sampling Distributions
Distributions of sample mean can be derived
using mathematical theorem.
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Central Limit Theorem If random samples of size
n are drawn from a population with mean µ and
standard deviation s, then, when n is large, the
sampling distribution of the sample mean
is approximately normally distributed, with mean
µ and standard deviation . The
approximation becomes more accurate as n becomes
large.
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Sampling Distribution of the Sample Mean

• A random sample of size n is selected from a
  population with mean µ and standard deviation s.
• The sampling distribution of always have
  mean µ and standard deviation .
• When n is large, the sampling distribution of the
  sample mean is approximately normally
  distributed, with mean µ and standard deviation
  .

The standard deviation of x-bar is sometimes
called the STANDARD ERROR (SE).
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How Large is Large?

• If the population is normal, then the sampling
  distribution of will also be normal for any
  sample size (no matter what value of n is).
• When the population is approximately symmetric,
  the distribution of becomes approximately
  normal for relatively small values of n.

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How Large is Large?

• When the population is skewed, the sample size n
  must be at least 30 before the sampling
  distribution of becomes approximately normal.

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Finding Probabilities for the Sample Mean

• If the sampling distribution of
• is normal or approximately normal, standardize or
  rescale the interval of interest in terms of
• Find the appropriate area using Table 3.

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Example
Example A random sample of size n 16 from a
normal distribution with µ 10 and s 8.
Calculate the probability of sample mean greater
than 12. P( gt12)?
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Example
A soda filling machine is to fill cans of soda.
Suppose that the fills are actually normally
distributed with mean 12.1 oz and standard
deviation of 0.2 oz. What is the probability that
the average fill for a 6-pack of soda is less
than 12 oz?
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Example
A soda filling machine is to fill cans of soda.
We do not know what kind of distribution the
fill has, but we know it has mean 12.1 oz and
standard deviation of 0.6 oz. What is the
probability that the average fill for a 30-pack
of soda is between 12 oz and 12.2 oz?
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Example

• In a market, the average money spent by each
• customer is 8.5 with standard deviation 2.
• Suppose that a random sample of 36 customers
• was selected.
• Within what limits would you expect the average
  spending of 36 customers to lie, with probability
  0.997?
• Calculate the probability P( lt7)?
• If the average spending of 36 customers is 7,
  would you think this unusual?

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Sample Proportion

• Suppose that we are interested in the proportion
  (p) of US people (population), who are still
  willing to fly following the events of Sept. 11,
  2001.
• We may take a random sample of n people to
  estimate the proportion p in the population (US
  people).
• Use x to represent the number of people who are
  willing fly in our sample.

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Sample Proportion

• The sample proportion, , which is
  simply a rescaling of the binomial random
  variable x, dividing it by n.

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Mean and Standard deviation of the Sample
Proportion

• A random sample of size n is selected from a
  binomial population with parameter p.
• The sampling distribution of the sample
  proportion, will have mean p and
  standard deviation .

The standard deviation of p-hat is sometimes
called the STANDARD ERROR (SE) of p-hat.
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Sampling Distribution of the Sample Proportion
From the Central Limit Theorem, the sampling
distribution of will also be
approximately normal, with mean p and standard
deviation .

• The approximation will be adequate if npgt5 and
  nqgt5.

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Finding Probabilities for the Sample Proportion

• If the sampling distribution of is normal
  or approximately normal, standardize or rescale
  the interval of interest in terms of
• Find the appropriate area using Table 3.

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Finding Probabilities for the Sample Proportion
Example A random sample of size n 100 from a
binomial population with p 0.4.
Calculate the probability of sample proportion
greater than 0.5. P( gt0.5)?
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Mean and Standard deviation of the Sample
Proportion

• A random sample of size n is selected from a
  binomial population with parameter p.
• The sampling distribution of the sample
  proportion, will have mean p and
  standard deviation .

The standard deviation of p-hat is sometimes
called the STANDARD ERROR (SE) of p-hat.
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Sampling Distribution of the Sample Proportion
From the Central Limit Theorem, the sampling
distribution of will also be
approximately normal, with mean p and standard
deviation .

• The approximation will be adequate if npgt5 and
  nqgt5.

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Example
The soda bottler claims that only 5 of the soda
cans are underfilled. A quality control
technician randomly samples 200 cans of soda.
What is the probability that more than 10 of
these 200 cans are underfilled?
n 200 S underfilled can p P(S) .05 q
.95 np 10 nq 190
OK to use the normal approximation
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Example
n 200 S underfilled can p P(S) .05 q
.95 np 10 nq 190
OK to use the normal approximation
This would be very unusual, if indeed p .05!
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Example

• A study shows that 46 people admit to
• overeating sweet foods when they are stressed.
• Now a random sample of n100 Americans is
• selected.
• Does the sample proportion of Americans who
  overeating sweet foods, have a normal
  distribution approximately?
• If so, what are its mean and standard deviation?

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Example

• A study shows that 46 people admit to
• overeating sweet foods when they are stressed.
• Now a random sample of n100 Americans is
• selected.
• What is the probability that the sample
  proportion lies within the interval 0.35 to 0.55?
  
• What might you conclude if the sample proportion
  were as small as 30?

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In-class Exercise

• You take a random sample of size 36 from a
• binomial distribution with parameter p0.4.
• Can the sampling distribution of be
  approximated by a normal distribution? Why?
• What is the mean and standard deviation of the
  sample mean ?
• Find the probability that the sample proportion
  is between 0.5 and 0.6.

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In-class Exercise

• You take a random sample of size 49 from a
  distribution with mean 53 and standard deviation
  21.
• What is the mean and standard deviation of the
  sample mean ?
• Find the probability that the sample mean is
  greater than 55.

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Key Concepts

• I. Statistics and Sampling Distributions
• Statistics take values varying from sample to
  sample, and hence are random variables, e.g.,
• is a random variable.
• The probability distributions of statistics,
  e.g., , are called sampling distributions.

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Key Concepts

• II. Sampling Distribution of the Sample Mean
• When samples of size n are drawn from a normal
  population with mean µ and variance
• s 2, the sample mean has a normal
  distribution with mean µ and variance s 2/n.

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Key Concepts

• 2. When samples of size n are drawn from a
  nonnormal population with mean µ and variance s
  2, the Central Limit Theorem ensures that the
  sample mean will have an approximately normal
  distribution with mean µ and variance s 2 /n when
  n is large than 30.
• 3. Probabilities involving the sample mean m can
  be calculated by standardizing the value of
  using

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Key Concepts

• III. Sampling Distribution of the Sample
  Proportion
• When samples of size n are drawn from a binomial
  population with parameter p, the sample
  proportion will have an approximately normal
  distribution with mean p and variance pq /n as
  long as np gt 5 and nq gt 5.
• 2. Probabilities involving the sample proportion
  can be calculated by standardizing the value
  using