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# chapter6c_prediction.pdf

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University of Waterloo

Statistics

STAT 231

Matthias Schonlau

Fall

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Chapter 6 - Prediction
Matthias Schonlau
Stat 231 Outline
• One sample prediction
• Regression prediction
– Example: Professor Cotton’s Marathon Prediction of a single obs
vs Averages
• We have made inferences for averages, and
for regression parameters α and β.
• The purpose of prediction is to make inference
for a single new observation.
• Single observations are more variable than
averages. Example: Driving to T oronto
• Give a confidence interval for the average driving
time from Waterloo to Toronto
– If you plan for the average driving time, you will arrive
late half the time and early the other half.
• You have tickets for the Toronto fringe theater
festival. Late arrivers are not seated. Give a
confidence interval for driving to Toronto for this
one drive.
• The confidence interval for one particular drive is
wider than that for the average. One sample Prediction
• Assume the one sample model
YR µ σ , R~G(0, )
• So far, we have learned how to
– estimateμ and σ
–(Confidence interval)inty of the estimation
– Hypothesis tests
• All this refers to already collected data.
• What if we want to predict a new data point? One Sample Prediction
• A single new data point drawn from this
model has expectation μ and standard
deviation σ
• Y is the random variable for the new point
0
Y0~G(μ,σ)
• We don’t know μ and σ, we only know their
estimates
• We also know the sampling distribution for μ
σ
µµ~ (, ) One Sample Prediction
• The error of the point estimate is given by
() −µ
0
• Note
()0 0−+−=+µ−Rµµ ( µ) ( )
• By independence:
2 2 2 1
Var(Y0 0) V+)r Y =+ =arµ+ σ
nn One Sample Prediction
1
Y0−µ+σ~ , 1 n
• There are two sources of variation:
– uncertainty for a single point
– uncertainty of the estimate for μ One sample prediction:
Hypothesis T est (T-test)
• The prediction variance is used in the usual
ways
• T-test
YY µµ
0 0 = ~ n 1
se()0 −µ 1
σ 1 +n One sample prediction:
Prediction Interval
1
µσ±+c*1
n
where c is the critical value form the t-distribution
with (n-1) d.f. Regression prediction
• Now assume the regression model:
Y =α+1 xR , R~G(0,σ)
• It is useful to rewrite this model, centering on
x:
Y =α+β−+ x( R ) , R~G(0,σ)
2
• where α12β − x Regression prediction:
Centred regression estimators
• For centered data,
x = 0
yx +=αβ α ˆ
• Estimates for α and β are independent
ˆ Sxy
β =
Sxx
• The distributions of the parameters are:
– This was shown earlier
α ~ G, σ , ~
n S xx Regression Prediction
• The estimator of the mean value at x newis
µ x +− α=β xx)
new new Regression Prediction
• By independence,
Var ( x( ) +)βVa() (ar )x x
new ( new )
=Var () x x Var β
( new ) ( )
2
=σ 1 +(xnew− )
nS xx
Regression prediction:
Distribution of estimator at x new
• By independence, the estimated value at x new
has the following distribution
2
1 (xnew )
µ(x new ~ x − ( +new ), nS
xx
• This predicted value is on the regression line.
• The variance refers to average Regression prediction:
Distribution of a single point at x
new
• The distribution of a single random draw at
xnewis :
Ynew xx (αβ+− ( new σ ), )
• The expected value of a single obs and the
average are the same
• The variance refers to a single observation Regression Prediction:
Combined variability
• By independence between Y and μ(x )
new new
2
Y x−G µσ ) ~ 0, 1++1 (xnew )
new new nS xx
Regression Prediction:
Hypothesis test
• Because the Y~G, the sampling distribution is T
• Hypothesis test:
Ynewµ() new
2 ~ n 2
1 (xnew )
σ 1++ nS
xx
• Subtracting 2 d.f.because α and β were
estimated Regression Prediction:
Confidence interval
2
ˆ 1 (xnew )
µˆ(xnew σ ++ nS
xx
Where c is the critical value from the T
distribution with (n-2) d.f. Regression

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