Lecture 22: Tue, Nov 26

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Lecture 22 Tue, Nov 26

• Announcements
• HW 8 posted, due Thurs, Dec 5th
• Todays Lecture
• Interval estimation of the population mean.
• Interpretation of confidence intervals.
• Computer simulations

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• The general form for a confidence
  interval is
• or, equivalently

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How to Use the Interval

• For inference, we often assume a particular value
  of mu.
• If this value is not contained within the
  confidence interval, then this is evidence
  against the hypothesized value.

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Size of the Interval

• What happens to the interval as n increases?
• When the confidence level increases?
• When the population SD increases?

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• Confidence Levels.

Z
0
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• Some commonly-used confidence levels and
  corresponding z-scores

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Interpreting the Interval

• Before the experiment, the probability that the
  confidence interval will cover the true parameter
  value is
• After the experiment, we say that, with .
  confidence, the interval covers
  the true parameter value.

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Incorrect Interpretation

• It is incorrect to say that the probability that
  lies within the interval is
  . .
• Why? Because is a fixed number, so that
  probability is either 0 or 1.

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Long-Run View of Confidence Intervals

• If we repeated our experiment over and over, and
  constructed 95 confidence intervals each time,
  we would expect about 95 of the intervals to
  cover the true value of
• JAVA Applets

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Information and Confidence Intervals

• Small interval ? more information.
• Larger interval ? less information.

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Inference using Confidence Intervals

• 1) Assume a particular value for mu.
• 2) Collect data construct confidence interval
• 3) If the hypothesized value of mu is not
  contained in the interval ? evidence that the
  value is incorrect.

C.I.
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Example

• A manufacturer claims that the lifetimes of their
  light bulbs are exponentially distributed with a
  mean of 100 hours.
• Each classroom at Penn contains n40 light bulbs
  from this manufacturer.
• In each classroom, observe lifetimes, and
  construct a 90 confidence interval.
• Approximately 90 of these confidence intervals
  will contain 100 hours.

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Exercise 10.20

• The following observations are the ages of a
  random sample of eight men in a bar. It is known
  that the ages are normally distributed with a
  standard deviation of 10. Determine the 95
  confidence interval estimate of the population
  mean. Interpret what the interval estimate tells
  you.
• 52, 68, 22, 35, 30, 56, 39, 48

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Exercise 10.24

• A statistics professor is investigating how many
  classes university students miss each semester.
  To answer this question, she took a random sample
  of 100 students and asked them how many classes
  they had missed in the previous semester.
• Estimate the mean number of classes missed by all
  students at the university. Use a 99 confidence
  level and assume that the population SD is known
  to be 2.2 classes.

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Components of a Confidence Interval
UCL
LCL
Width on each side
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Example

• Let be a random sample from a
  population with (unknown) mean and standard
  deviation .
• The 95 confidence interval for the mean is given
  by 0,10, and is centered around the sample
  mean. Find the 90 confidence interval for the
  mean .
• Assume the sample size in (a) is n100. What is
  the value of .

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Confidence Intervals fora Proportion

• XBinomial(n,p), p unknown
• Given X, how to construct a confidence interval
  for p?
• A use the sampling distribution of

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• For npgt5 and n(1-p)gt5, is approximately
  normal
• So

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• After some algebra, we get
• Plugging in for p, we get a confidence
  interval

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Example

• 100 Florida voters were polled on Election Day
  2000 on their choice for president. 60 of the
  respondents said they voted for Bush, and 40 said
  they voted for Gore.
• a) Find a 95 confidence interval for the true
  proportion of Florida voters that favored Bush
  over Gore.

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Quick Quiz

• Which of the following four components can the
  experimenter control?
• Sample mean
• Confidence level
• Population variance
• Sample size

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Quick Quiz

• Describe what happens to the width of the
  confidence interval when each of the following
  happens.
• Confidence level increases
• Confidence level decreases
• Population variance increases
• Population standard deviation decreases
• Sample size increases
• Sample size decreases
• Sample mean increases
• Sample mean decreases