Sampling Distribution Models

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Chapter 18

• Sampling Distribution Models

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Demonstration

• Observe in class SPSS demonstration related to
  sampling distribution models

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Demonstration Summary

• First, we examined the distribution of state
  appropriations for education given the entire
  population of U.S. states.
• Our findings indicated that the distribution of
  state spending on education was skewed to the
  right, with a mean (m) of 1,272,969,120.00 and
  standard deviation (s) of 1,567,930,688.096

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Demonstration Summary

• Next we randomly selected 30 states to be
  included in our sample. Analysis of this sample
  indicated that again the distribution of spending
  on education was skewed to the right however the
  mean of the sample ( )was 1,410,710,766.67
  with a standard deviation (s) of
  1,941,673,134.577.

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Demonstration Summary

• We then repeated the random sampling process to
  get a new sample of thirty states. We noticed
  that this new sample also had a distribution that
  was skewed to the right however, the mean and
  standard deviation of this sample differed. The
  results were 1,126,093,266.67 and
  1,781,298,838.439 respectively.
• Did we do something wrong?

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Demonstration Summary

• We then examined 100 different random samples of
  size thirty and determined that each sample had a
  slightly different mean and standard deviation
  due to sampling variability (i.e. different
  combinations of states were included in each of
  our samples).
• When we went to create a histogram for our
  collection of sample means, we discovered
  something pretty amazing that distribution
  looked very much like a normal model even though
  the distribution of state appropriations from our
  original population was skewed to the right.

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Sampling Distribution

• A listing of all the values that a sample mean
  can take on and how often those values can occur
  is called the sampling distribution of a sample
  mean.
• This histogram of sample means depicts the
  sampling distribution of the sample mean.
• Like any other distribution, a sampling
  distribution of the sample mean has a shape,
  center, and measure of variability (i.e. spread)
• This distribution can be interpreted as the
  probability distribution of sample means.
• Under certain conditions this sampling
  distribution will approximate the normal model
  regardless of the shape of the distribution for
  the original variable from the population.

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Simulating the Sampling Distribution of a Mean

• We can use simulation to get a sense as to what
  the sampling distribution of the sample mean
  might look like
• Lets start with a simulation of 10,000 tosses of
  a die. A histogram of the results is

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Means Averaging More Dice

• Looking at the average of two dice after a
  simulation of 10,000 tosses

• The average of 5 dice after a simulation of
  10,000 tosses looks like

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Means What the Simulations Show

• As the sample size (number of dice) gets larger,
  each sample average is more likely to be closer
  to the population mean.
• So, we see the shape continuing to tighten around
  3.5
• And, it probably does not shock you that the
  sampling distribution of this mean becomes
  Normal.

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The Central Limit Theorem (CLT)

• The mean of a random sample has a sampling
  distribution whose shape can be approximated by a
  Normal model. The larger the sample, the better
  the approximation will be.
• The CLT is surprising and a bit weird
• Not only does the histogram of the sample means
  get closer and closer to the Normal model as the
  sample size grows, but this is true regardless of
  the shape of the population distribution.
• All we need is for the observations to be
  independent and collected with randomization.

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Conditions Required for the CLT

1. Random Sampling Condition The data values must
   be sampled randomly or the concept of a sampling
   distribution makes no sense.
2. Independence Assumption Impossible to know for
   sure, instead use the 10 condition the sample
   size, n, is no more than 10 of the population.

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But Which Normal?

• Recall that normal models are described by their
  means and standard deviations.
• The mean of all sample means is the population
  mean m. That is to say, the sampling
  distribution of the mean has a mean m .
• The standard deviation of all sample means is
  . That is to say, the sampling
  distribution of the mean has a standard deviation
  .

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The Sampling Distribution Model for a Mean

• When a random sample is drawn from any population
  with mean m and standard deviation s , its sample
  mean has a sampling distribution with the
  same mean m but whose standard deviation is
• (we write ).

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The Sampling Distribution Model for a Mean
(continued)

• No matter what population (whether it has a
  distribution that is symmetric, uniform, or
  skewed to the right or left) the random sample
  comes from, the shape of the sampling
  distribution is approximately Normal as long as
  the sample size is large enough. The larger the
  sample used, the more closely the Normal
  approximates the sampling distribution for the
  mean.

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Sampling Distributions for Proportions

• The Central limit theorem does not apply only to
  sample means
• Can make the same conclusions about the shape,
  center and variability about the sample
  proportions.
• The sample proportion is denoted by and is
  equal to the number of individuals in the sample
  in the category of interest, divided by the total
  sample size (n).

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What About the Sampling Distribution Model for a
Proportion

• Provided that the sampled values are independent
  and the sample size is large enough, the sampling
  distribution of (sample proportion) is modeled
  by a Normal model with
• Mean
• Standard deviation
• Where p is the probability of success (i.e.
  observation falls into the specific group of the
  categorical variable that you are interested in).
  q is the probability of failure.

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Necessary Conditions When Working with Proportions

• Two assumptions
• The sampled values must be independent of each
  other
• The sample size, n, must be large enough
• Check the following corresponding conditions
• 10 Conditions sample size must be no larger
  than 10 percent of the population
• Success/Failure Condition sample size must be
  large enough such that np and nq are at least
  10. In other words we need to expect at least 10
  success and 10 failures to have enough data for a
  sound conclusion.

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Standard Error

• The standard deviations of our Normal models are
  as follows
• For proportions For means
• When we dont know p or s, were stuck, right?

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Standard Error (cont)

• Nope. We will use sample statistics to estimate
  these population parameters.
• For a sample proportion, the standard error is
• For the sample mean, the standard error is
• When we estimate the standard deviation of a
  sampling distribution using statistics found from
  the data, the estimate is called a standard
  error.

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Watch out for small samples from skewed
populations

1. If the original population is not itself normally
   distributed, here is a common guideline For
   samples of size n greater than 30, the
   distribution of the sample means can be
   approximated reasonably well by a normal model.
   The approximation gets better as the sample size,
   n, becomes larger.
2. If the original population is itself normally
   distributed, then the sample means will be
   normally distributed for any sample size n (not
   just values of n larger than 30).

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Applications of the Central Limit Theorem - 1

• In the 2001 ACT, students had a mean score of
  21.3 with a standard deviation of 6.0. Assume
  that the scores are normally distributed.
• If 60 students are randomly selected, find the
  probability that they have a mean score greater
  than 23.5.

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Applications of the Central Limit Theorem - 2

• A national study found that 44 of college
  students engage in binge drinking (5 drinks at a
  sitting for men, 4 for women). Use the
  68-95-99.7 Rule to describe the sampling
  distribution model for the proportion of students
  in a randomly selected group of 200 college
  students who engage in binge drinking. Do you
  think the appropriate conditions are met?

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Example 3

• Carbon monoxide emissions for a certain kind of
  car vary with mean 2.9 g/m and standard deviation
  0.4 g/m. A company has 80 of these cars in its
  fleet.
• Estimate the probability that the mean CO level
  for the companys fleet is between 3.0 and 3.1
  g/m.
• There is only a 5 percent chance that the fleets
  mean CO level is greater than what value?

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Example 4

• Just before a referendum on a school budget, a
  local newspaper polls 400 voters in an attempt to
  predict whether the budget will pass. Suppose
  that the budget actually has the support of 52
  of the voters. Whats the probability the
  newspapers sample will lead them to predict
  defeat? Be sure to verify that the assumptions
  and conditions necessary for your analysis are
  met.

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Assignment

• Read Chapter 18 Again!
• Try the following exercises from Ch. 18
• 1, 3, 7, 9, 17, 21, 23, 25, 27, 33, 37
• Work through the ActivStats assignments for
  Chapter 18 for additional practice.