Discrete distribution word problems

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• Discrete distribution word problems
• Probabilities specific values, gt, lt, lt, gt,
• Means, variances
• Computing normal probabilities and inverse
  values
• Pr(Xlty) when y is above and below the mean of X
• Pr(y1ltXlty2) when y1 and y2 are
• both above the mean of X
• both below the mean of X
• on opposite sides of the mean of X
• Central Limit Theorem
• Sum version
• Average version

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Apply Central Limit Theorem to Estimates of
Proportions
Source gallup.com Suppose this is based on a
poll of 100 people
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This uses the averageversion of the CLT.
Twolectures ago, we appliedthe sum version of
the CLT to the binomial distribution.
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• Suppose true p is 0.40.
• If survey is conducted again on 49 people, whats
  the probability of seeing 38 to 42 favorable
  responses?
• Pr( 0.38 lt P lt 0.42)
• Pr(0.38-0.40)/sqrt(0.620.38/49) lt Z lt
  (0.42-0.40)/sqrt(0.620.38/49)
• Pr(-0.29 lt Z lt 0.29) 2Pr(Zlt-0.29) 0.77

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• Chapter 8
• In the previous example, the random quantity was
  the estimator.
• Examples of estimators
• Sample mean X (X1Xn)/n
• Sample variance (X1-X)2(Xn-X)2/(n-1)
• Sample median midpoint of the data
• Regression line .
• ESTIMATORS CALCULATE STATISTISTICS FROM DATA

If data are random, then the estimators are
randomtoo.
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• Central limit theorem tells us that the
  estimators X and P have normal distributions as n
  gets large
• X N(m,s2/n) where m and s are the mean and
  standard deviation of the random variables that
  go into X.
• P N(p,p(1-p)/n) where p is true proportion of
  yeses

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• Two ways of ways to evaluate estimators
• Bias Collect the same size data set over and
  over. Difference between the average of the
  estimator and the true value is the bias of the
  estimator.
• VarianceCollect the same size data set over and
  over. Variability is a measure of how closely
  each estimate agrees.

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Distribution of abiased estimator
Distribution of an unbiased estimator
Example The median is a biased estimate of the
true meanwhen the distribution is skewed.
Bias inaccuarcy Variance imprecision
True value
Distribution of a lessvariable estimator
Distribution of amore variable estimator
True value