Confidence Intervals

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Confidence Intervals
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Body Temperature

• 97.8 98.0 98.2 98.2
• 98.2 98.6 98.8 98.8
• 99.2 99.4
• Calculate the sample mean
• Assume standard deviation of population is 0.53

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Point Estimator

• Sample Mean
• Do we expect our sample mean to be the true
  population mean?
• How much confidence do we have that we are close?

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Sampling Distribution Facts

• Mean of the sampling distribution is the same as
  the unknown mean of the population
• The standard deviation of x-bar is
• The Central Limit Theorem tells us that x-bar has
  a distribution that is close to Normal as long as
  n 30

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Statistical Estimation

• Use x-bar to estimate µ.
• x-bar is an unbiased estimator of µ but will
  rarely actually be µ.
• In repeated samples, the values of x-bar follow a
  Normal distribution
• The 68-95-99.7 rule for Normal distributions
  allows us to construct interval estimates at 68,
  95 and 99.7 (actually any value of our choice)

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95 Confidence Interval

• ( x-bar - 2 , x-bar 2 )
• This interval will capture the true mean, µ, in
  about 95 of all possible samples of size n.

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• Interpreting a 95 Confidence Interval
• Once we have constructed our interval estimate
  there are only two possibilities
• The interval contains the true mean, µ
• - or -
• 2. Our sample produced an x-bar that is not
  within two standard deviations of the true mean,
  µ. (Only 5 of samples of size n would give such
  inaccurate results.)

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• We cant know if we have one of these unlucky
  results.
• We can say that
• We are 95 confident that our interval includes
  the true mean for the population.

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Not a probability statement! We used a method
that gives correct results 95 of the time.
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Interval Estimators Estimate margin of
error Confidence intervals can be constructed at
any level. 95 is common but you can use 99,
90 or even 80
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• 90 net 95 net

µ
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Conditions for Constructing a Confidence Interval
for a Population Mean

• The data come from an SRS from the population of
  interest (not blocked, clustered or stratified)
• The sampling distribution of x-bar is
  approximately Normal (when? Two cases)
• Individual observations are independent (sampling
  without replacement, the population size must be
  at least 10 times the sample size)

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• These three conditions must be met in order for a
  confidence interval to be valid!!!
• Always check that these conditions are met!!!
• On AP Test You must show that the conditions
  are met before constructing a confidence interval

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• Constructing a Confidence Interval for a mean at
  any level
• Estimate the mean with x-bar
• z where z is the critical value.
• Find z in table. (1 C)/2
• -or-
• InvNorm((1C)/2) standard normal

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General Confidence Interval

• Estimate Margin of Error
• Estimate Critical Value of z (or t) Standard
  Error
• Standard Error standard deviation of the sample
  mean or

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Inference Toolbox p. 548

• To construct a confidence interval
• Parameter. Identify the population of interest
  and the parameter you want to draw conclusions
  about.
• Conditions. Choose the appropriate inference
  procedure. Verify conditions for using it.

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Inference Toolbox p. 548

• To construct a confidence interval (continued)
• Calculations. If the conditions are met, carry
  out the inference procedure
• CI estimate margin of error
• 4. Interpretation. Interpret your results in
  the context of the problem.

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How Confidence Intervals Behave

• Higher confidence implies a wider interval.
• Increasing the sample size reduces the margin or
  error for any fixed confidence level. Because we
  take the square root of n, we must take 4 times
  as many observations in order to cut the margin
  of error in half.

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Determining Sample Size

• Sample size for a desired margin of error m.
• m z s/vn
• Example We want a margin of error of 1 unit at
  a 95 confidence interval
• 1 1.965/vn
• Solve for n
• Always round n UP

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Population standard deviation is known

• This is hardly ever the case
• However,
• It must be assumed at this point in our study
• Sometimes you can reasonably estimate the
  population standard deviation when you know the
  range of data. Divide the range by 6.

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Sample Size Alone

• Determines the margin of error
• The size of the population has no effect.

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Cautions

• The data must be from an SRS.
• Different methods apply for any data other than
  from an SRS.
• There is NO correct method for inference from
  data haphazardly collected with bias of unknown
  size.
• Outliers can distort results. (Correct, remove.
  If not, use other procedures)

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Cautions

• The shape of the population distribution matters.
  If the sample size is small and the population
  is not normal, check for skewness and other signs
  of Non-Normality.
• You must know the standard deviation of the
  population.
• The margin of error in a confidence interval
  covers only random sampling errors. (not
  undercoverage, nonresponse errors)

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Meaning of a 95 Confidence Interval

• We are 95 confident that the true mean lies in
  our interval. That is, these numbers were
  calculated by a method that gives correct results
  in 95 of all possible samples.
• We cannot say that there is a 95 probability
  that we have captured the true mean.

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Practice

• Textbook p. 542
• Exercise 10.1
• Exercise 10.2
• Exercise 10.3
• Exercise 10.4