Statistical Sciences 2244A/B Chapter Notes - Chapter 23: Squared Deviations From The Mean, Dependent And Independent Variables, Test Statistic
22 May 2018
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Department
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Stats 2244
Chapter 23
CHAPTER 23.1
Conditions for regression inference
- Because the conclusions of inference always concern some population, the conditions describe
the population and how the data are produced from it
- The slope b and intercept a of the least-squares line are statistics.
- That is, we calculated them from the sample data.
- These statistics would take somewhat different values if we repeated the analysis with different
oysters.
- To do inference, think of a and b as estimates of unknown parameters that describe the
population of all oysters.
- Conditions for regression inference:
o We have n pairs of observations on an explanatory variable x and a response variable y
o Goal: study or predict the behavior of y for given values of x
o For any fixed value of x, the response y varies according to a normal distribution
o Repeated responses y are independent of eachother
o The mean response μy has a straight-line relationship with x given by a population
regression line
o The slope and intercept are unknown parameters.
o The standard deviation of y (call it σ) is the same for all values of x. The value of σ is
unknown.
o There are thus three population parameters that we must estimate from the
data: , , and σ.
- The population regression line μy = +x says that the mean response μy moves along a straight
line as the explanatory variable x changes. We can’t observe the population regression line. The
values of y that we do observe vary about their means according to a Normal distribution.
- The standard deviation σ determines whether the points fall close to the population regression
line (small σ) or are widely scattered (large σ).
- In the image:
o the line in the figure is the population regression line.
o The mean of the response y moves along this line as the explanatory variable x takes
different values.
o The Normal curves are the distribution of y at different fixed values of x.
o All the curves have the same σ, so the variability of y is the same for all values of x.
o You should check the conditions for inference when you do inference about regression
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Document Summary
Because the conclusions of inference always concern some population, the conditions describe the population and how the data are produced from it. The slope b and intercept a of the least-squares line are statistics. That is, we calculated them from the sample data. These statistics would take somewhat different values if we repeated the analysis with different oysters. To do inference, think of a and b as estimates of unknown parameters that describe the population of all oysters. The value of is unknown: there are thus three population parameters that we must estimate from the data: (cid:573), (cid:574), and . The population regression line y = (cid:573)+(cid:574)x says that the mean response y moves along a straight line as the explanatory variable x changes. The values of y that we do observe vary about their means according to a normal distribution.
