Write

1
Write

• Describe the relationship between confidence
  level and the size of your interval.
• That is, as each one goes up, what happens to the
  others?
• Does this relationship make sense to you? Why or
  why not?

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Confidence Interval for the Mean When Var(X)
Unkown

• Wednesday 30 January 2008

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The Confidence Interval

• Trades accuracy for certainty.
• Rests on two numbers
• Sample mean
• Some margin of error based on a probability
  distribution and the standard deviation

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The Problem

• We need to know something about the normal
  distribution to use Mondays method.
• If we dont have the standard deviation, things
  get sketchy.
• So, we use whats called the t-distribution.

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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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Trivia

• The derivation of the t-distribution was first
  published in 1908 by William Sealy Gosset, while
  he worked at a Guinness brewery in Dublin.
• He was not allowed to publish under his own name,
  so the paper was written under the pseudonym
  Student.

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Finding Critical Values for t-dist.

• Youre going to use a chart.
• It looks like this.

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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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Computing a Confidence Interval for Unknown SD

• 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.
• Look familiar?

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The Only Tricky Part

• Finding the critical value for student-t depends
  on sample size (n), which is new.
• The degrees of freedom, keep in mind, is equal to
  n-1.
• So determine your confidence, subtract 1 from
  the sample size, and go to the table.

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The Margin of Error for Unknown SD
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For Example

• I measure the temperature on 8 days in January,
  and find a sample mean for the high of 28
  degrees, with a sample standard deviation of 5.4
  degrees.
• Construct a 95 confidence interval for the mean.

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Lets try a few

• 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.
• For the sake of comparison, compute the interval
  if you had asked 500 people.
• 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.
• Again, compute if n 600.

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A few with data

• Monthly incomes for 14 randomly-selected people
  with degrees in economics (dollars)
• 3450 3596 3366 3455 3151 2727 3283 3527 3407 4036
  4083 2946 3023 3806
• Monthly incomes for 10 people with degrees in
  biology
• 2148 1978 2093 2091 2282 2223 2276 2207 2285 2159
• Can you make a statement at the 90 confidence
  level comparing the two populations?

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Four more.

• In a random sample of 70 bolts, the mean length
  was 1.25 inches with an SD of 0.01 inches.
• You took a random sample of 12 toasters, and
  found a mean price of 61.12 with an SD of
  24.62.
• You sample 23 people visiting an emergency room,
  and found a mean wait time of 24 minutes with an
  SD of 11 minutes.
• In a random sample of 17 shoppers at Cumbys, the
  mean amount spent was 18.13 with an SD of 6.05.
  

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The Student-t Distribution

• Its shape depends on degrees of freedom.
• For n lt 30, use t.
• For unknown SD, use t and the margin of error
  given by