Lecture 5 Chapter 3. Confidence Intervals and Hypothesis Tests

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Lecture 5Chapter 3. Confidence Intervals and
Hypothesis Tests
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• An Important Property of the Normal Distribution
• It is always true that 95 of the probability
  lies within 1.96 standard deviations of the mean.

• In popular scientific wisdom, the usual range
  of a population lies within 2 standard deviations
  of the mean.

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• Now in an actual experiment, we will usually
  collect a single sample of data from a
  population.
• Example (artificial data generated by me!)
• A random sample of 16 students taking MAS1401
  sit an IQ test, giving the following results
• 124 114 116 119 123 124 137 115
  118 120 105 115 111 134 132 140
• Sample mean 121.69
• Sample standard deviation 9.79
• What can these data tell you about the mean IQ
  of all students taking MAS1401?
• This is a good example of using a sample to try
  and answer a question about a population.

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• It is reasonable to assume that the IQs come
  from a Normal distribution.
• However the mean, µ, and the standard deviation,
  s, are liable to be different from those of the
  general population.
• You are interested in µ, the population mean IQ
  for MAS1401.
• You can estimate its value using the sample
  mean, 121.69.
• This isnt particularly useful however, unless
  you can give some idea of how accurate the
  estimate is.
• What would be better would be to construct a
• confidence interval for µ.

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• So how did our sample mean value,121.69, arise?
• Well by the Central Limit Theorem, it came from a
  Normal distribution with mean µ, and standard
  deviation s/vn s/v16 s/4.
• Using the slide entitled An important property
  of the Normal distribution (above), we can be
  95 confident that population mean µ lies within
• 1.96 x (s/4) of 121.69.
• Suppose you were to know s, then you could use
  this to construct a 95 confidence interval for
  µ. In fact, for this example, I can tell you that
  s 9.

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• So, straight away, you can construct a 95
  confidence interval for µ as
• sample mean (1.96 x s/vn)
• which is
• 121.69 (1.96 x 9/4)
• i.e. the range of numbers
• from 121.69 (1.96 x 9/4) up
  to 121.69 (1.96 x 9/4)
• which gives the 95 confidence interval
• (117.28, 126.10)
• for µ.