CONFIDENCE INTERVALS

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CONFIDENCE INTERVALS

• 2/21/06

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Outline

• Return / Review Exams
• Confidence Intervals when ? is known
• Margin of Error
• Break
• Confidence Intervals when ? is unknown
• In class project

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A flighty example

• Say a biologist wants to determine the home range
  of a particular species of bird.
• From all the population of this species, the
  biologist collects a sample of 100 birds and
  attaches a radio transmitter to the leg of each
  bird.
• After monitoring the position of the birds for 1
  month, the biologist finds that the mean home
  range of the sample is 23.1 km.
• Is this the exact value of the population?

NO!
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Point Estimation

• The question we can ask is, Well how close is
  23.1 to the population mean?
• This value (23.1) is called the point estimate,
  it is the value we will use to estimate the
  population value
• This is the Best Guess of the value of the
  population parameter, given your sample
  information
• The way we will answer this question is to
  generate a range of values for which we can be
  reasonably confident that the population value
  falls
• This process is called interval estimation
• The resulting range of values is called a
  confidence interval.

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Point Estimate Example

• The signing bonus for 30 new players in the NFL
  are used to estimate the mean bonus for all new
  players. The sample mean is 130,000 with a
  standard deviation of 25,500. What is the point
  estimate of the mean signing bonus for all new
  NFL players?
• Answer the sample mean is 130,000 so this is
  your point estimate of the population mean, ?

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Returning to our flighty example

• The biologist has a point estimate of 23.1 km,
  and can guess that the true population is
  probably between 20 and 26.
• While the biologist may be confident that the
  true range is between 20 and 26, she may be even
  more confident that with a range of 15-30.
• Thus, the wider the interval estimation, the
  greater the confidence that we have that the
  interval contains the population mean.

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Confidence Intervals

• Now, it is possible and preferable to be more
  quantitative about how we go about determining
  the interval (rather than just guessing).
• This is what well do for the rest of class
• So, a confidence interval provides a range of
  numbers along with the percentage confidence that
  the parameter lies within
• A 95 confidence interval means that 95 of
  similarly constructed intervals will contain the
  population parameter
• Note also that although there are many different
  CIs that we could construct, in practice the 90,
  95, and 99 CI are used most often.

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Computing Confidence Intervals Sampling Error

• Last class, we saw that whenever we use a sample
  to estimate a population characteristics, we are
  going to have some amount of sampling error.
• Sometimes, we will overestimate the true value
• Sometimes, we will underestimate the true value

Sample 2
Sample 3
Sample 4
Sample 1
True Value
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Standard Error

• But, we also know (from the central limit
  theorem) that if our sample size is sufficiently
  large (n gt 30), the sampling distributions of the
  sample means will be approximately normal.
• Thus, 95 of the sample means will fall between
  2 standard deviations from the mean of the
  population
• The standard deviation of the sample means
  Standard Error, or the degree to which particular
  means of samples are typically in error as
  estimates of the mean of the population

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What does this mean?

• When we collect a sample of data, we can be
  reasonably certain that the true population value
  falls within 2 standard deviations (plus or
  minus) of our sample mean!
• Bottom line If you have?X and add and subtract
  about 2 standard deviations from it, this is 95
  confidence interval

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Elements of a Confidence Interval
Sample statistic (point estimate)
Confidence interval
Confidence limit (lower)
Confidence limit (upper)
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Calculating confidence intervals

• Two methods
• When the standard deviation of the population is
  known
• When the standard deviation of the population is
  unknown
• When will we already know the standard deviation
  of the population?
• Well, most of the time this value is unknown
• However, sometimes a previous study (or group of
  studies) has established the standard deviation
  for a population.
• What else do we need?
• Set level of confidence

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Level of Confidence and Alpha level

• What determines the width of our confidence
  interval?
• We do! Before we start calculating the CI, we
  must determine our level of confidence
• This is another way of saying How comfortable am
  I with being wrong
• Alpha (?) indicates the probability that our
  confidence interval does not include the
  population parameter
• The level of confidence we set determines our
  alpha
• Level of Confidence 1 - ?
• Thus, if we set a confidence of 95, our alpha
  will be (1-95) or 5
• So, if we have a 95 CI, this means that we will
  have a 5 chance that our CI will not include the
  true value.

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Confidence Interval Procedure when ? is known.
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Dividing ? in half

• Note that when we set alpha, we need to consider
  the fact that sometimes our point estimate will
  underestimate the population value and sometimes
  will overestimate the true value.
• So this means that for a 95 CI, we want 2.5
  chance of making an overestimation and 2.5
  chance of making an underestimation.

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Common Levels of Confidence

• Commonly used confidence levels are 90, 95, and
  99

Confidence Coefficient,
Normalz value,
Confidence Level
1 - ?
Z ?/2
1.28 1.645 1.96 2.33 2.58 3.08 3.27
.80 .90 .95 .98 .99 .998 .999
80 90 95 98 99 99.8 99.9
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Confidence Depends on Interval (z)
?X? ? Z??x

?X
?
?1.65??x ?2.575??x
?-2.575??x ?-1.65??x
?1.96??x
?-1.96??x
90 CI
95 CI
99 CI
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Guidelines for using this procedure

• For small sample sizes, (n lt 15) z curve can only
  be used with the variable under consideration is
  normally distributed
• For moderate sample sizes, (15 lt n lt 30) z curve
  can be used unless the data contains extreme
  values or the sample is not normally distributed
• For large sample sizes (n gt 30), the z curve can
  be used no matter what the distribution of the
  variable under consideration.

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Factors Affecting Interval Width
Intervals Extend from?X - Z??X to?X Z??X

• 1. Data Dispersion
• Measured by ?
• 2. Sample Size
• ??X ? / ?n
• 3. Level of Confidence (1 - ?)
• Affects Z

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Lets return to our flighty example

• Our biologist found a point estimate of 23.1 km.
• Say the biologist knows that the standard
  deviation of the population of birds is 4.7km
• To find the 95CI,
• 23.1 1.96 (4.7/?100)
• 23.1 .9212
• 95 CI 22.18 to 24.02

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More examples on board
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Does interval estimation work?

• The best way to test whether this works is to run
  a simulation.
• http//www.ruf.rice.edu/7Elane/stat_sim/conf_inte
  rval/index.html

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Margin of Error

• In our equation,

This part of equation is the margin of error or
E
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Margin of Error

• Sometimes, we will have in mind a specific margin
  of error before we start our study.
• For example, some political pollsters say I want
  to determine the job approval for candidate A
  with a margin of error of some value
• When we have a predetermined margin of error, we
  can determine the sample size needed to get the
  estimate of the population value, within that
  margin

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Margin of Error Equation

• E (Z?/2 x ?) / ?n
• Through some algebra, we get
• n (Z?/2 x ?)/n2
• Lets do an example

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Margin of Error Example

• On Board

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Class project