The Central Limit Theorem

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Section 6-5

• The Central Limit Theorem

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The Central Limit Theorem

• Suppose the random variable x has a
• distribution (that may or may not be normal)
  with mean ? and standard dev ?.

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• Then, the distribution of the sample
• means, based on a random sample
• of size n, will have a distribution that
• approaches a normal distribution.

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• The mean of all sample means
• will equal the population mean ? and
• the standard deviation
• will equal as n increases without limit.

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• ? is often called the standard error of
  the mean
• ?This (CLT) usually applies when n gt 30
• the approximation gets better and better
• as the sample size increases.
• ?If the original population IS normally
• distributed, then the sample means will be
• normally distributed for ANY sample size!

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Example 1

• Assume that mens weights are normally
  distributed with a men given by µ 172 lb and a
  standard deviation given by s 29 lb (based on
  data from the National Health Survey).
• a. If 4 men are randomly selected, find the
  probability that they have a mean weight between
  160 lb and 180 lb.

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• ?P( 160 lt x lt 180)
• b. Why can the central limit theorem be used in
  part (a), even though the sample size does not
  exceed 30?

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Example 2

• For women aged 18-24, systolic blood pressure
  (in mm Hg) are normally distributed with a mean
  of 114.8 and a standard deviation of 13.1.
  Hypertension is commonly defined as a systolic
  blood pressure above 140.

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• If a woman between the ages of 18 and 24 is
  randomly selected, find the probability that her
  systolic blood pressure is above 140.
• P(x gt 140)

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• If 4 women in that age bracket are randomly
  selected, find the probability that their mean
  systolic blood pressure is greater than 140.
• P(x gt 140)

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• c. If a physician is given a report stating that
  4 women have a mean systolic blood pressure
  below 140, can she conclude that none of the
  women have hypertension (with a blood pressure
  greater than 140)?