Transcript Document

Chapter 14 Inference for Regression
• Objectives:
– Review how to make a scatterplot to show
relationship between explanatory and response
variables.
– Use a calculator to find the correlation and leastsquare regression line.
– Recognize the regression setting for a line.
– Recognize the type of inference needed in a
particular regression setting.
– Inspect data for non-linear relationship, influential
observations, skewed data, or nonconstant variation.
– Explain meaning of slope, intercept, and standard
error.
– Determine tests and confidence intervals for ß.
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14.1 Inference About the Model
• A scatter plot shows a linear relationship between quantitative
explanatory variable x and quantitative response variable y.
– Use least square regression line to predict y.
• Look at form, direction, and strength of relationship as well as outliers or
other deviations.
– Numerical summary – correlation describes strength and direction of
relationship. It will explain how much of the variation is due to the
explanatory value.
– Mathematical model – LSRL ŷ = a + bx
• Conditions for Regression Inference
– For any fixed value of x, the response y varies according to a normal
distribution. Repeated responses y are independent of each other.
– The mean response μy has a straight line relationship with x:
True line regression: μy = α + βx where α and β are unknown parameters.
– The standard deviation of y (call it σ) is the same for all values of x. The
value of σ is unknown.
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14.1 Inference for Regression (cont.)
• Standard Error about the line
S = n 1 2  ( y  y)2
• Confidence Interval for Regression Slope
Estimate + t*SEb
s
SEb=  ( x  x )
t*is upper (1-c)/2 critical value with a n-2 degree of
freedom
• Statistic Test of H0: β=0 t statistic where
2
t=
b
SEb
calculator linear regression t test is LinRegTTest
(Remember regression analysis usually give a two sided
P value so if you need one sided just divide the P value
by 2.)
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14.2 Predictions and Conditions
• To estimate the mean response for a LSR we use a confidence
interval.
μy = α + βx*..
• A level of confidence interval for the mean response μy when x takes
on a value of x* is
• The standard error is
• To estimate the individual response we use a prediction interval. A
prediction interval estimates a single random response y rather than a
parameter like μy . The response is not a fixed number
• The level of confidence C prediction interval for a single observation
on y when x takes on the value x* is
• The standard error is
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14.2 Predictions and Conditions (cont.)
1.
2.
3.
4.
Conditions
Observations are independent
The true relationship is linear.
The standard deviation of the response about the true
line is the same everywhere.
The response varies normally about the true
regression line.
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