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Inference for Regression - Model, Predictions, and Conditions

Learn how to make inferences about the regression model, make predictions, and check the conditions for regression inference. Understand the plot and interpretation, explanatory and response variables, correlation, LSRL, and more.

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Inference for Regression - Model, Predictions, and Conditions

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  1. Chapter 14Inference for Regression AP Statistics 14.1 – Inference about the Model 14.2 – Predictions and Conditions

  2. Two Quantitative Variables • Plot and Interpret • Explanatory Variable and Response Variable • FSDD • Numerical Summary • Correlation (r) – describes strength and direction • Mathematical Model • LSRL for predicting

  3. 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 • Parameters of Interest: • The standard deviation of y (call it ) is the same for all values of x. The value of is unknown  t-procedures! • Degrees of Freedom: n – 2

  4. Conditions for Regression Inference (Cont’d) • Look at residuals: • residual = Actual – Predicted • The true relationship is linear • Response varies Normally about the True regression line • To estimate , use standard error about the line (s)

  5. Inference • Unknown parameters: • a and b are unbiased estimators of the least squares regression line for the true intercept and slope , respectively • There are n residuals, one for each data point. The residuals from a LSRL always have mean zero. This simplifies their standard error.

  6. Standard Error about the Line • Two variables gives: n – 2 df (not n – 1) • Call the sample standard deviation (s) a standard error to emphasize that it is estimated from data • Calculator will calculate s! Thank you TI!

  7. t-procedures (n - 2 df) • CI’s for the regression slope standard error of the LSRL slopeb is: • Testing hypothesis of No linear relationship • x does not predict y r = 0

  8. What is the equation of the LSRL? • Estimate the parameters • In your opinion, is the LSRL an appropriate model for the data? Would you be willing to predict a students height, if you knew that his arm span is 76 inches?

  9. Construct a 95% CI for mean • increase in IQ for each additional • peak in crying

  10. Scatter Plot and LSRL?Perform a Test of Significance

  11. Checking the Regression Conditions • All observations are Independent • There is a true LINEAR relationship • The Standard Deviation of the response variable (y) about the true line is the Same everywhere • The response (y) varies Normally about the true regression line * Verifying Conditions uses the Residuals!

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