Confidence Intervals

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Confidence Intervals
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Overview

• To frame our discussion, consider

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Outline

• Foundations
• Large Sample
• Small Sample

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Confidence Interval

• A 100(1-a) confidence interval for the mean µ of
  a normal population is
• means that if we were to take samples of size n
  repeatedly and compute a 90 confidence level
  confidence interval for the population mean from
  each sample of size n, the long-run fraction of
  intervals that contain the population mean would
  converge to 90.

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Confidence Interval

• a is the desired level of confidence.

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Sample Size

• The general formula for sample size, n, necessary
  to ensure an interval width is

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Large Sample Interval for µ

• We would like to establish to some degree of
  certainty that a parameter falls with a given
  range.
• For large n, n30,

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• Typically the standard deviation of the
  population is unknown. When this is the case we
  use the sample standard deviation, s, in the
  place of s, when the sample size is large (ngt40).

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Error

• In using the mean of a large sample to estimate
  the mean of a population, we want to assert with
  probability 1-a the maximum error.
• Note this formula can be used to determine the
  required sample size if you fix acceptable error.

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Population Proportion
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t distribution

• Given the mean of a random sample of size n from
  a normal distribution with mean µ,
• Has a probability distribution called the t
  distribution with n-1 degrees of freedom.

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Properties
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Students t distribution

• Applied when working with normal or nearly normal
  populations.
• Distribution is similar to normal. Shape is a
  function of the number of degrees of freedom.
• Degrees of freedom (n) is n-k, where k is the
  number of population parameters that must be
  estimated.

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• The techniques discussed require that s be known
  or s can be approximated with s. Hence, n30.
• If it is reasonable to assume we are sampling
  from a normal distribution, we can use the
  t-distribution

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Variance

• Let X1, X2, , Xn be a random sample from N(µ,
  s2). The r.v.
• Has a chi-squared (c2) probability distribution
  with n-1 degrees of freedom (df).

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• Like the t-distribution, the actual c2
  distribution is based on the degrees of freedom
  (typically dfn-1).

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Example

• Determine a and b such that P(altc2ltb).95 where
  df12.

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Solution
Notice the c2 distribution is not symmetric.
Table p. 755
Value b from table using .025 and df12. Value a
requires from table using .975 and df12.
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Sampling Distribution s2

• The c2 random variable is describes the s2
  sampling distribution. In fact,

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Confidence Intervals
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Problem 1

• A random sample of 100 programmers in the Boston
  area revealed a mean monthly salary of 1100 with
  a standard deviation of 120. With what degree
  of confidence can we assert that the average
  monthly salary is between 1000 and 1200.

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Problem 2

• QA wants to determine the average time it takes
  an individual to prepare for a code inspection.
  She wants to be able to assert with 99
  confidence that the mean of her sample is within
  2 minutes. Historical data suggests that s3.5
  minutes. What is the sample size required to
  establish this accuracy?

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Problem 3

• A computer scientist, trying to optimize system
  performance, collected data on the time, in
  microseconds, between requests for a process
  service. She recorded 50 observations with a mean
  of 11795 and a standard deviation of 14054. What
  are the endpoints of the 95 confidence interval
  of the true mean?

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Problem 4

• Inspecting LCD screen prior to system
  configuration, QC engineers detect 22, 23, 26,
  20, 24, and 29 defects in six production runs
  each of size 200. Construct a 98 confidence
  interval for the true number of defects per
  production run of size 200.