Chapter 5: Sampling Distributions

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Chapter 5 Sampling Distributions
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1. Sampling Distribution

• Until now, we have only talked about population
  distributions.
• Population Distribution (of a variable)
• The distribution of all the members of the
  population.
• Sampling Distribution
• This is not the distribution of the sample.
• The sampling distribution is the distribution of
  a statistic.
• If we take many, many samples and get the
  statistic for each of those samples, the
  distribution of all those statistics is the
  sampling distribution.
• We will most often be interested in the sampling
  distribution of the sample mean or the sample
  proportion.

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1. Sampling Distribution

• Example Suppose the proportion of those who
  agree with a particular UN policy is 0.53.
• Population distribution?
• Suppose we randomly sample 1000 individuals and
  ask them if they agree with the UN policy.
• What is the distribution of the sample
  proportion?
• Does this distribution differ from the population
  distribution?

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1. Sampling Distribution

• Example Scores on an Intelligence Scale for the
  20 to 34 age group are normally distributed with
  mean 110 and standard deviation 25.
• Population distribution?
• Suppose we sample 50 individuals between 20 and
  34 and obtain the mean and standard deviation of
  that sample.
• What is the distribution of the sample mean?
• Does this distribution differ from the population
  distribution?

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2. Sampling Distributions for Counts and
Proportions

• Eg) Toss a fair coin 20 times.
• The number of heads is a random variable X.
• The true proportion of heads is 0.5
• If 11 of the tosses are heads, X 11. And the
  sample proportion is
• Count a random variable X above is a count of
  the occurrences of some outcome (head) in a fixed
  number of observations (20).
• Sample proportion if the number of observations
  is n
• Note We have called the population proportion p
  and sample proportion p, but the book uses p as
  the population proportion, and as the
  sample proportion.

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2. Sampling Distributions for Counts and
Proportions

• The distribution of the count X of success (head)
  has a binomial distribution.
• Rules for the binomial setting
• There are a fixed number n of observations.
• The n observations are all independent.
• Each observation falls into one of just two
  categories, which for convenience we call
  success and failure.
• The probability of a success, call it p, is the
  same for each observation.

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2. Sampling Distributions for Counts and
Proportions

• If a variable X is the count of successes out of
  n, then X has the binomial distribution. XB(n,
  p).
• n is the number of observations
• p is the probability of success of any one
  observation.
• The mean and standard deviation of count X B(n,
  p) is
• So the mean and standard deviation of
  is

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2. Sampling Distributions for Counts and
Proportions

• Example
• Suppose the population is all members of
  fraternities and sororities on campus. The
  population size is 12,000.
• We take a random sample of 1000 members and ask
  Have you had five or more drinks at one time
  during the last week? We will call the answer
  yes a success.
• Also, suppose that the true number of people who
  drank more than five drinks at one time last week
  is 5456.
• The number of people out of our sample who
  answered yes, X, was 423.
• What is the approximate sampling distribution of
  X?

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x 586
x 300

• Example Counts
• What does this mean?

x 482
Suppose we take many, many samples (of size 1000)
x 328
x 274
x 444
and so forth
Then we find the sample count for each sample.
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2. Sampling Distributions for Counts and
Proportions

• Example Counts
• Then the distribution of all of these sample
  counts, xs (300, 586, 482, 444, 328, 274, etc.)
  is B(1000, (5456/12000))
• i.e., Binomial(1000, .455)

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2. Sampling Distributions for Counts and
Proportions

• Normal Approximation for counts and proportions
• Eg) Probability histogram of the sample
  proportion based on a binomial count with n
  2500 and p 0.6. The distribution is very close
  to normal.
• In fact, both the count X and the sample
  proportion are approximately normal in large
  samples.

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2. Sampling Distributions for Counts and
Proportions

• Counts
• The distribution of the sample counts, or
  sampling distribution of X is approximately
  normal

when n (the sample size) is large. As a rule of
thumb, use this approximation for values of n and
p that satisfy np 10 and n(1 p) 10.
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2. Sampling Distributions for Counts and
Proportions

• Proportions
• The distribution of the sample proportions, or
  sampling distribution of p is approximately
  normal

when n (the sample size) is large. As a rule of
thumb, use this approximation for values of n and
p that satisfy np 10 and n(1 p) 10.
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2. Sampling Distributions for Counts and
Proportions

• Example Proportions
• Suppose a large department store chain is
  considering opening a new store in a town of
  15,000 people.
• Further, suppose that 11,541 of the people in the
  town are willing to patronize the store, but this
  is unknown to the department store chain
  managers.
• Before making the decision to open the new store,
  a market survey is conducted.
• 200 people are randomly selected and interviewed.
  Of the 200 interviewed, 162 say they would
  patronize the new store.

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2. Sampling Distributions for Counts and
Proportions

• Example Proportions
• What is the population proportion p?
• 11,541/15,000 0.77
• What is the sample proportion p?
• 162/200 0.81
• What is the approximate sampling distribution (of
  the sample proportion)?

What does this mean?
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p 0.82
p 0.73

• Example Proportions
• What does this mean?

p 0.82
Suppose we take many, many samples (of size 200)
p 0.78
p 0.74
p 0.76
and so forth
Then we find the sample proportion for each
sample.
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2. Sampling Distributions for Counts and
Proportions

• Example Proportions

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2. Sampling Distributions for Counts and
Proportions

• Example Proportions
• The managers didnt know the true proportion so
  they took a sample.
• As we have seen, the samples vary.
• However, because we know how the sampling
  distribution behaves, we can get a good idea of
  how close we are to the true proportion.
• This is why we have looked so much at the normal
  distribution.
• Mathematically, the normal distribution is the
  sampling distribution of the sample proportion,
  and, as we will see, the sampling distribution of
  the sample mean as well.

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3. Sampling Distribution for Sample Mean

• If the population distribution is normal, then so
  is the distribution of the sample mean.
• If a population has the N(?,?²) distribution,
  then the sample mean of n independent
  observations has the N(?,?²/n) distribution.
• Eg) the height X of a single randomly chosen
  young woman varies according to the N(64.5, 2.52)
  distribution. If a medical study asked the height
  of an SRS of 100 young women, what would be the
  sampling distribution of the sample mean height?

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3. Sampling Distribution for Sample Mean

• Sample Mean
• Let be the mean of an SRS of size n from a
  population having mean ? and standard deviation
  ?. The mean and standard deviation of are

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3. Sampling Distribution for Sample Mean

• The Central Limit Theorem states that for any
  population with mean m and standard deviation s,
  the sampling distribution of the sample mean is
  approximately normal when n is large

How large a sample size n is need depends on the
population distribution. More observations are
required if the shape of the population is far
from normal.
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3. Sampling Distribution for Sample Mean

• The Central Limit Theorem
• The central limit theorem is a very powerful tool
  in statistics.
• Remember, the central limit theorem works for any
  distribution.

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3. Sampling Distribution for Sample Mean

• Example Sample Mean
• There has been some concern that young children
  are spending too much time watching television.
• A study in Columbia, South Carolina recorded the
  number of cartoon shows watched per child from
  700 a.m. to 100 p.m. on a particular Saturday
  morning by 28 different children.
• The results were as follows
• 2, 2, 1, 3, 3, 5, 7, 5, 3, 8, 1, 4, 0, 4, 2, 0,
  4, 2, 7, 3, 6, 1, 3, 5, 6, 4, 4, 4. (Adapted from
  Intro. to Statistics, Milton, McTeer and Corbet,
  1997)
• Suppose the true average for all of South
  Carolina is 3.4 with a standard deviation of 2.1.

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3. Sampling Distribution for Sample Mean

• Example Sample Mean
• What is the population mean?
• What is the sample mean?
• What is the sampling distribution (of the sample
  mean)?

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mean 3.7

• Example Sample Mean

mean 4.1
Suppose we take many, many samples (of size 28)
mean 3.5
mean 3.2
mean 2.6
and so forth
Then we find the sample mean for each sample.
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3. Sampling Distribution for Sample Mean

• Example Sample Mean

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3. Sampling Distribution for Sample Mean

• Example Sample Mean
• Similar to the example for sample proportions,
  the sampling distribution of the sample means
  follow a normal distribution.
• This allows us to determine with some certainty
  how likely our sample mean is to be near the true
  population mean.
• In reality, we dont have the luxury of obtaining
  many, many samples. We can only assume we do and
  say our sample is one of those many.