P. STATISTICS LESSON 14

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P. STATISTICSLESSON 14 2( DAY 2)

• PREDICTIONS AND CONDITIONS

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ESSENTIAL QUESTIONS What are the conditions that
must be in place in order to make predictions?

• Objective
• To create confidence intervals and prediction
  intervals for a single observation.

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Predictions and conditions

• One of the most common reasons to fit a line to
  data is to predict the response to a particular
  value of the explanatory variable.
• y -0.0127 0.0180(5) 0.077
• Do you want to predict the BAC of one individual
  student who drink 5 beers and all students who
  drink 5 beers.

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Predictions and Conditions cont.

• The margin of error is different for the two
  kinds of prediction. Individual students who
  drink 5 beers dont all have the same BAC. So we
  need a larger margin of error to pin down one
  students who have 5 beers.

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Prediction and confidence intervals

• To estimate the mean response, we use a
  confidence interval. It is an ordinary interval
  for the parameter
• µy a ßx
• The regression model says that µy is the mean of
  response y when x has the value x. It is a
  fixed number whose value we dont know.

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Prediction interval

• To estimate an individual response y, we use a
  prediction interval. A prediction interval
  estimates a single random response y rather than
  a parameter like µy. The response y is not a
  fixed number. If we took more observations with
  x x, we would get different responses

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Confidence intervals for regression response

• A level C confidence interval for the mean
  response µ when x takes the value x is
• y t SEµ
• The standard error SE is
• SE s v 1/n (x - x)2/ ?(x - x)2
• The sum runs over all the observations on the
  explanatory variable x.

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Prediction intervals for regression response

• A level C prediction interval for a single
  observation on y when x takes the value x is
• y t SEy
• The standard error for prediction SEy is
• SEy sv 1 1/n (x - x)2/ ?(x - x)2
• In both recipes, t is the upper (1-C)/2 critical
  value of the t distribution with
• n 2 degrees of freedom.

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Example 14.7

• Predicting Blood Alcohol
• Page 798
• Look at minitab.

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Checking the regression conditions

• If the scatterplot doesnt show a roughly linear
  pattern, the fitted line may be almost useless.
• The observations are independent.
• In particular, repeated observations on the
  same individual are not allowed.
• The true relationship is linear.
• we almost never see a perfect straight-line
  relationship in our data.

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Checking the regression conditions continued

• The standard deviation of the response about the
  true line is the same everywhere. Look at the
  scatterplot again. The scatter of the data
  points about the line should be roughly the same
  over the entire range of the data.
• The response varies normally about the true
  regression line. We cant observe the true
  regression line. We can observe the
  least-squares line and the residual, which show
  the variation of the response about the fitted
  line.