AP STATISTICSAP STATISTICS

LESSON 14 – 1 LESSON 14 – 1 ( DAY 1 )( DAY 1 )

INFERENCE ABOUT THE INFERENCE ABOUT THE MODELMODEL

ESSENTIAL QUESTION: ESSENTIAL QUESTION: What is regression inference and What is regression inference and how is it used?how is it used?

Objectives:Objectives:• To find regression inference.To find regression inference.• To find standard errors for regression To find standard errors for regression

lines.lines.• To create confidence intervals for To create confidence intervals for

regression slope.regression slope.

Inference About the ModelInference About the Model

When a scatterplot shows a linear When a scatterplot shows a linear relationship between a quantitative relationship between a quantitative explanatory variable x and a explanatory variable x and a quantitative response variable y, we can quantitative response variable y, we can use the least-squares line fitted to the use the least-squares line fitted to the data to predict y for a given value of x.data to predict y for a given value of x.

Example 14.1 Page 781Example 14.1 Page 781Crying and IQCrying and IQ

• Plot and interpret.Plot and interpret.

• Numerical summaryNumerical summary

• Mathematical model. Mathematical model.

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We are interested in We are interested in predicting the predicting the response from response from information about the information about the explanatory variable. explanatory variable. So we find the least So we find the least square regression line square regression line for predicting IQ from for predicting IQ from crying.crying. y = a + bxy = a + bx^

The Regression ModelThe Regression Model

We use the notation y to remind ourselves We use the notation y to remind ourselves that the regression line gives predictions of that the regression line gives predictions of IQ.IQ.

The slope b and intercept a of the least-The slope b and intercept a of the least-squares line of are statistics. That is we squares line of are statistics. That is we calculate them from the sample data.calculate them from the sample data.

To do formal inference, we think of a and b as To do formal inference, we think of a and b as estimates of unknown parameters.estimates of unknown parameters.

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Conditions for Regression Conditions for Regression InferenceInference

We have n observations on an explanatory We have n observations on an explanatory variable x and a response variable y. Our variable x and a response variable y. Our goal is to study or predict the behavior of y for goal is to study or predict the behavior of y for given values of x.given values of x.

• For any fixed value of x, the response y varies For any fixed value of x, the response y varies according to a normal distribution. Repeated according to a normal distribution. Repeated responses y are independent of each other.responses y are independent of each other.

Conditions of Regression Conditions of Regression (continued…)(continued…)

• The mean response The mean response μμy y has a straight-line has a straight-line relationship with x:relationship with x:

μμy y = = αα + +ββxx

The slope The slope ββ and intercept and intercept αα are unknown are unknown parameters.parameters.

• The standard deviation of y (call it The standard deviation of y (call it σσ ) is the same ) is the same for all values of x. The value of for all values of x. The value of σσ is unknown. is unknown.

The Heart of the The Heart of the Regression ModelRegression Model

The heart of this model is that there is an The heart of this model is that there is an “on “on the average”the average” straight-line relationship straight-line relationship between y and x. The true regression line between y and x. The true regression line μμy y

= = αα + +ββx says that the mean response x says that the mean response μμyy moves along a straight line as the moves along a straight line as the explanatory variable x changes.explanatory variable x changes.

The mean of the response y moves along this The mean of the response y moves along this line as the explanatory variable x takes line as the explanatory variable x takes different valuesdifferent values

Inference Inference

The first step in inference is to estimate The first step in inference is to estimate the unknown parameters the unknown parameters αα, , ββ, and , and σσ. .

The slope b is an unbiased estimator of The slope b is an unbiased estimator of the true slope the true slope ββ, and the intercept a of , and the intercept a of the least-squares line is an unbiased the least-squares line is an unbiased estimator of the true intercept estimator of the true intercept αα. .

Example 14.2 Page 784Example 14.2 Page 784Slope and InterceptSlope and Intercept

A slope is a rate of change. A slope is a rate of change.

The true slope The true slope ββ says how much higher says how much higher average IQ is for children with one more average IQ is for children with one more peak in their crying measurement.peak in their crying measurement.

We need the intercept We need the intercept αα to draw the to draw the

line, but it has no statistical meaning. line, but it has no statistical meaning.

Example 14.2 Example 14.2 (continued…)(continued…)

The standard deviation The standard deviation σσ, which describes the , which describes the variability of the response y about the true variability of the response y about the true regression line. regression line.

The least-squares line estimates the true The least-squares line estimates the true regression line. Recall that the residuals are regression line. Recall that the residuals are the vertical deviations of the data points from the vertical deviations of the data points from the least-squares line:the least-squares line:

Residual = observed y – predicted y = y - yResidual = observed y – predicted y = y - y

Standard Error About the Standard Error About the Least-Squares LineLeast-Squares Line

We call this sample standard deviation a standard error We call this sample standard deviation a standard error to emphasize that it is estimated from data.to emphasize that it is estimated from data.

The standard error about the line is The standard error about the line is

s = √ 1/(n – 2)∑ residuals = √ 1/(n – 2)∑ residual22

s = s = √√ 1/(n – 2)∑ (y – y) 1/(n – 2)∑ (y – y)22

Use s to estimate the unknown Use s to estimate the unknown σσ in the regression in the regression

model.model.

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