The Central Limit Theorem

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Math 1107 Introduction to Statistics

• Lecture 11
• The Central Limit Theorem

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MATH 1107 The Central Limit Theorem

• Often we cannot analyze a population directlywe
  have to take a sample. What are some of the
  reasons we sample?

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MATH 1107 The Central Limit Theorem
After we take a sample, in order to make
inferences onto the population, we have to assume
the data is normally distributed. What if our
population is not normal? Do we have a problem?
http//www.ruf.rice.edu/lane/stat_sim/sampling_di
st/index.html
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MATH 1107 The Central Limit Theorem

• Important concepts to remember about the Central
  Limit Theorem
• The distribution of sample means will, as the
  sample size increases approach a normal
  distribution
• The mean of all sample means is the population
  mean
• The std of all sample means is the std of the
  population/the SQRT of the sample size
• If the population is NOT normally distributed,
  sample sizes must be greater than 30 to assume
  normality
• If the population IS normally distributed,
  samples can be of any size to assume normality
  (although greater than 30 is always preferred).

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MATH 1107 The Central Limit Theorem
Example of Application (Page 262) If a Gondola
can only carry 12 people or 2004 lbs safely,
there is an inherent assumption that each
individual will weigh 167 lbs or less. Men weigh
on average 172 lbs, with a std of 29 lbs.
Assume that any selection of 12 people is a
sample taken from an infinite population. What
is the probability that 12 randomly selected men
will have a mean that is greater than 167 lbs?
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MATH 1107 The Central Limit Theorem
Because we assume that weight is normally
distributed (it almost always is), we can
comfortably use a sample less than 30. We can
also assume that the mean of our samples will be
the same as our population mean, and that the std
of our sample is the same as the population
std/SQRT of the sample size? 29/SQRT(12) or
8.372 Now, we can calculate a Z-score Z
167-172/8.372. This equals -.60. Or 73.
What is the interpretation of this figure?
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MATH 1107 The Central Limit Theorem
Example of Application (Page 265) Assume that
the population of human body temperatures has a
mean of 98.6F. And, the std is .62F. If a
sample size of n106 is selected, find the
probability of getting a mean of 98.2F or
lower. Here, we dont know how the population is
distributed, but because the sample size is
greater than 30, it does not matterwe can assume
the distribution of the sample is normal.
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MATH 1107 The Central Limit Theorem
Again, we assume that the sample means will be
the same as the population mean (98.6) and the
std of the samples is the same as the std of the
population/SQRT of the sample size
(.62/SQRT(106)). Now, we can calculate a
Z-score Z98.2-98.6/.06022 -6.64 What does
this mean?