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 least square 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 μ y has a straight line relationship with x: True line regression: μ y = 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*SE b SE b =  (

x s



x

) 2 t*is upper (1-c)/2 critical value with a n-2 degree of freedom • Statistic Test of H 0 : β=0 t statistic where t=

b SE b

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.) 3

14.2 Predictions and Conditions

• To estimate the mean response for a LSR we use a confidence interval .

μ y = α + βx*.

.

• A level of confidence interval for the mean response μ y on a value of x* is when x takes • 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 μ y . 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 4

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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