Sampling Distributions

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Presentation 2

• Sampling Distributions

• Sampling Distribution of sample proportions
• Sampling Distribution of sample means

2
Statistics VS Parameters

• Statistic is a numerical value computed from a
  sample.
• Parameter is a numerical value associated with
  a population.
• Essentially, we would like to know the parameter.
  But in most cases it is hard to know the
  parameter since the population is too large.
• So we have to estimate the parameter by some
  proper statistics computed from the sample.

3
Some Notation

• p population proportion
• sample proportion
• µ population mean
• sample mean
• s standard deviation
• s sample standard deviation

4
Sampling Distribution of the Sample Proportion

• Situation 1 A survey is undertaken to
  determine the proportion of PSU students who
  engage in under-age drinking. The survey asks 200
  random under-age students (assume no problems
  with bias). Suppose the true population
  proportion of those who drink is 60.
• Thus, p 0.6 and is the proportion in
  the sample who drink.

5
Repeated Samples

• Imagine repeating this survey many times, and
  each time we record the sample proportion of
  those who have engaged in under-age drinking.
  What would the sampling distribution of look
  like?

Sample (n200) Sample Proportion
1 1
2 2
3 3
4 4
5 5

150,000 150,000
is a random variable assigning a value to
each sample!
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Histogram of for 150 000 samples.
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Sampling Distribution of

• Let X be the number of respondents who say they
  engage in under age drinking.
• X is binomial with n 200 and p 0.6.
• So, we can calculate the probability of X for
  each possible outcome (0-200). The PDF is
  plotted below

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Sampling Distribution of

• Since X Bin (n 200, p 0.6), the sampling
  distribution of is the same as that of the
  binomial distribution divided by n.
• Therefore we have

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Sampling Distribution of - Cont.

• Using the Normal approximation to the binomial
  distribution we have that the sampling
  distribution of is approximately Normal
  with mean p and std. dev.
• i.e.
• The conditions for this approximation to be valid
  are
• The sample selected from the population is
  random.
• The sample must be large enough, np and n(1-p)
  MUST be greater than 5, and should be greater
  than 10.

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Example

• Recent studies have shown that about 20 of
  American adults fit the medical definition of
  being obese.
• A large medical clinic would like to estimate
  what percent of their patients are obese, so they
  take a random sample of 100 patients and find
  that 18 percent are obese.
• Suppose in truth, the same percentage holds for
  the patients of the medical clinic as for the
  general population, 20.
• Give notation and the numerical value for the
  following.

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Problem - Cont.

1. The population proportion of obese patients in
   the medical clinic
2. The proportion of obese patients in the sample of
   100 patients
3. The mean of the sampling distribution of
4. The standard deviation of the sampling
   distribution of
5. The variance of the sampling distribution of

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B. Sampling Distribution of the Sample Mean

• Situation 2 The mean height of women age 20 to
  30, X , is normally distributed (bell-shaped)
  with a mean of 65 inches and a standard deviation
  of 3 inches. i.e.
• X N(65,9)
• A random sample of 200 women was taken and
  the sample mean recorded.
• Now IMAGINE taking MANY samples of size 200 from
  the population of women. For each sample we
  record the . What is the sampling
  distribution of ?

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Histograms for the Distribution of X and X -Bar
Original Population of Women X height of random
woman
Distribution of Sample Means X-bar mean
of random sample of size 200.
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Normal Data

• Consider a Normal random variable X with mean µ
  and standard deviation s,
• X N( µ , s2 ).
• The sampling distribution of the sample mean of X
  for a sample of size n is Normal with
• i.e.

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Skewed or Non-Normal Data
Situation 3 In a college survey, students were
asked to report the number of cds they own.
Clearly CDs is a right skewed data set. Suppose
our population looked something like this, let us
take repeated samples from this population and
see what the sample mean looks like.
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Suppose we take repeated samples of size n 4,
8, 16, 32
n 4
n 8
n 32
n 16
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Statistics From Skewed Data

• Using that CD sample as the population,
• µ 87.6, s 87.8
• The sample means from the previous slide had the
  following summary statistics

Sample Size Mean of X-bar Std. Dev. of X-bar
n 4 86.6 43.2
n 8 86.8 30.9
n 16 86.7 21.9
n 32 86.6 15.6
Note that the mean remains constant, and the
std. deviation decreases as the sample size
increases!
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Central Limit Theorem

• For non-normal data coming from a population with
  mean µ and standard deviation s the sampling
  distribution of the sample mean is approximately
  normal with
• Conditions The above is true if the sample size
  is large enough, usually n gt 30 is sufficient.

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What next?

• We have shown that both the sampling distribution
  of the sample proportion, and the sampling
  distribution of the sample mean are both normal
  under certain conditions.
• Now we can use what we know about normal
  distributions to make conclusions about and
  !
• In the following we will see how to use the
  values of the statistics (p-hat, x-bar) to make
  inferences about the parameters (p, µ).

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Exercise 1

• The population proportion is 0.30. Consider the
  following questions.
• Find the sampling distribution of p-hat for each
  of the following sample sizes n100, n200,
  n1000
• What is the probability that a sample proportion
  will be within .04 of the population proportion
  for each of these sample sizes?
• What is the advantage of larger sample size?

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Exercise 2

• A certain antibiotic in known to cure 85 of
  strep bacteria infections. A scientist wants to
  make sure the drug does not lose its potency over
  time. He treats 100 strep patients with a 1 year
  old supply of the antibiotic. Let be the
  proportion of individuals who are cured.
• ASSUME the drug has NOT lost potency, answer the
  following questions
• What is the sampling distribution of ? Draw a
  picture
• If we repeated this study many times we would
  expect 95 of to fall within what interval?
• What is the probability that more than 90 in the
  sample are cured?

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Exercise 3

• A newspaper conducts a poll to determine the
  proportion of adults who favor a certain
  candidate. They ask a random sample of 800
  people whether or not they favor that candidate
  (Assume no bias!). Suppose the true proportion
  of adults who favor the candidate is 58.
• The newspaper records the sample proportion who
  favor the candidate. What is the sampling
  distribution of the sample proportion? Draw a
  picture of its PDF (center it correctly and
  include the appropriate scale).
• What is the probability that the newspaper would
  have recorded a sample proportion greater than
  62?
• What is the probability that less than 50 of the
  newspaper respondents would support this
  candidate?
• What is the probability that a randomly selected
  individual favors this candidate?

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Exercise 4

• Suppose the number of calories FIT students
  consume in a day is normally distributed with
  mean 2000 and standard deviation 300.
• About 95 of PSU students have a daily caloric
  intake between what two values?
• What is the probability that a randomly selected
  individual consumed between 1800 and 2100
  calories yesterday?
• Suppose I take a random sample of 36 students and
  recorded the number of calories each consumed on
  a given day. Describe the sampling distribution
  of the sample mean.
• Draw a picture of the sampling distribution of
  the sample mean (center it correctly and include
  the appropriate scale).
• If I take a sample of size 36 from the student
  body, what is the probability that the sample
  mean will be less than 2050?

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Exercise 5

• Assume the length of trout living in the
  Susquehanna River is normally distributed with
  mean of 14 inches and standard deviation of 2
  inches. A random sample of 16 trout is taken from
  the river.
• What is the sampling distribution of the average
  trout length (i) in a sample of size 16 (ii) in a
  sample of size 100?
• What happens to the sampling distribution of the
  sample mean as the sample size increases? (Draw a
  picture)
• What is the probability that a random sample of
  16 trout will provide a sample mean within one in
  of the population mean?
• What is the probability that a random sample of
  100 trout will provide a sample mean within one
  in of the population mean?
• What is the advantage of a larger sample size
  when one is attempting to estimate the population
  mean?