Write

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Write

• Which of these predictions are you more confident
  will be true? Why?
• Tomorrow, the high temperature will be between 23
  and 26 degrees.
• Tomorrow the high temperature will be between 18
  and 35 degrees.

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Review Confidence Intervals

• Thursday 14 May 2008

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The problem with statistics

• If were dealing with a statistic whose value
  depends on which sample we choose
• And if the parameter is essentially unknowable
• What good is the statistic? We dont know how
  close it is to the parameter!!!

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The Big Question

• What good is the statistic? We dont know how
  close it is to the parameter!!!
• The values for a statistic, taken many times,
  will be normally distributed.
• And of course, we know a lot about the normal
  distribution.

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Statistics and the Parameter

• As we re-sample and re-sample, the values we get
  for the statistic will be distributed normally
  around the population mean.

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Confidence Interval for a Statistic

• Tomorrow, the high temperature will be between 23
  and 26 degrees.
• Tomorrow the high temperature will be between 18
  and 35 degrees.
• The more confident we want to be, the larger we
  have to make our interval.
• We trade precision for certainty.

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Building a Confidence Interval

• Find the point estimate for the parameter.
• Choose a level of confidence.
• Get a critical value for this level of
  confidence.
• Determine the margin of error.
• Add and subtract this margin of error to the
  mean, and thats your interval!!!

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CI for the Mean when n gt 30 and variance is known

• Take the mean, add and subtract your margin of
  error.
• 90 z 1.645
• 95 z 1.960
• 99 z 2.575

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THE CATCH!!!!!

• This method, obviously, depends on knowing the
  standard deviation.
• Usually, you dont.

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Choosing a Distribution

• Normal (z critical value)
• n gt 30
• Standard deviation known
• Student t (t-table critical value)
• n lt 31 or
• Standard deviation unknown
• Always use the real standard deviation (not the
  sample standard dev.) if you have it.

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The Student-t Distribution

• Actually a series of distributions.
• Its shape depends on degrees of freedom.
• Equal to the number of observations minus the
  number of other stats you compute.
• Since were going to compute the mean, itll
  always be equal to n-1.

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Computing a Confidence Interval for Unknown
Variance

• Point estimate of the mean.
• Sample standard deviation.
• Choose confidence level.
• Determine critical value.
• Construct margin of error.
• Build your interval by adding and subtracting.

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The Margin of Error for Unknown SD
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Examples

• In a sample of 10 randomly-selected adults, you
  find that the mean amount of garbage produced per
  day is 4.3 pounds, with a sample SD of 1.2
  pounds.
• In a random sample of 12 adults, you find that
  the mean recycled waste per day is 1.2 pounds,
  with a sample SD of 0.3 pound.

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An Example (Bock Velleman DeVeaux)

• Sea Fan
• Recently under attack by aspergillosis.
• What suffer from this disease?

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An Example (Bock Velleman DeVeaux)

• Scientists sample 104 sea fans off the Las Redes
  Reef in Maxico.
• 54 show evidence of aspergillosis.

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Working with p-hat

• We can compute a point estimate for our
  proportion of infected coral.
• Also called p-hat
• We can use this to compute a sample standard
  deviation.
• Also called standard error.

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Distribution of P-Hat
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95 (or so) of values
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Stating Conclusions

• 51.9 of all sea fans are infected
• It is probably true that 51.9 of sea fans are
  infected.
• Between 42.1 and 61.7 of sea fans are
  infected.
• We can be 95 confident that between 42.1 and
  61.7 of sea fans are infected.

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A One-Proportion Z-Interval
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Lets Try One
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The Problem With s
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The Chi Square Distribution

• The standard deviation distributes itself along
  the chi-square probability distribution.
• Its asymmetrical.
• So well need 2 critical values.

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?2 Degrees of Freedom
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A Gift!
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Critical Values
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Confidence Interval for Variance
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Confidence Interval for Variance

• n is sample size
• s is sample standard deviation
• X2 are the critical values for the appropriate
  level of confidence and degrees of freedom

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Critical Values

• Youre going to need to divide confidence level
  by 2, and do some subtraction.
• This is again because of the asymmetry of the
  chi-square distribution.

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Confidence Interval for Variance

• Notice left and right.
• This makes sense because the right number will be
  larger, so as a factor in the denominator, it
  results in a smaller number (hence the less than).

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For Example

• A sample of 28 students finds that they brush
  their teeth on average 15.1 times per week, with
  a sample standard deviation of 2.7 toothbrushings.