1. AP STATISTICS LESSON 14 – 1 ( DAY 1 ) INFERENCE ABOUT THE MODEL

2. ESSENTIAL QUESTION: What is regression inference and how is it used? Objectives: To find regression inference.To find regression inference. To find standard errors for regression lines.To find standard errors for regression lines. To create confidence intervals for regression slope.To create confidence intervals for regression slope.

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

4. Example 14.1 Page 781 Crying and IQ Plot and interpret.Plot and interpret. Numerical summaryNumerical summary Mathematical model.Mathematical model. ^ We are interested in predicting the response from information about the explanatory variable. So we find the least square regression line for predicting IQ from crying. y = a + bx ^

5. The Regression Model We use the notation y to remind ourselves that the regression line gives predictions of IQ. We use the notation y to remind ourselves that the regression line gives predictions of IQ. The slope b and intercept a of the least- squares line of are statistics. That is we calculate them from the sample data. The slope b and intercept a of the least- squares line of are statistics. That is we calculate them from the sample data. To do formal inference, we think of a and b as estimates of unknown parameters. To do formal inference, we think of a and b as estimates of unknown parameters. ^

6. Conditions for Regression Inference We have n observations on an explanatory variable x and a response variable y. Our goal is to study or predict the behavior of y for given values of x. We have n observations on an explanatory variable x and a response variable y. Our goal is to study or predict the behavior of y for given values of x. For any fixed value of x, the response y varies according to a normal distribution. Repeated responses y are independent of each other.For any fixed value of x, the response y varies according to a normal distribution. Repeated responses y are independent of each other.

7. Conditions of Regression (continued…) The mean response μ y has a straight-line relationship with x:The mean response μ y has a straight-line relationship with x: μ y = α +βx The slope β and intercept α 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.The standard deviation of y (call it σ ) is the same for all values of x. The value of σ is unknown.

8. The Heart of the Regression Model The heart of this model is that there is an “on the average” straight-line relationship between y and x. The true regression line μ y = α +βx says that the mean response μ y moves along a straight line as the explanatory variable x changes. The heart of this model is that there is an “on the average” straight-line relationship between y and x. The true regression line μ y = α +βx says that the mean response μ y moves along a straight line as the explanatory variable x changes. The mean of the response y moves along this line as the explanatory variable x takes different values The mean of the response y moves along this line as the explanatory variable x takes different values

10. Inference The first step in inference is to estimate the unknown parameters α, β, and σ. The first step in inference is to estimate the unknown parameters α, β, and σ. The slope b is an unbiased estimator of the true slope β, and the intercept a of the least-squares line is an unbiased estimator of the true intercept α. The slope b is an unbiased estimator of the true slope β, and the intercept a of the least-squares line is an unbiased estimator of the true intercept α.

11. Example 14.2 Page 784 Slope and Intercept A slope is a rate of change. A slope is a rate of change. The true slope β says how much higher average IQ is for children with one more peak in their crying measurement. The true slope β says how much higher average IQ is for children with one more peak in their crying measurement. We need the intercept α to draw the line, but it has no statistical meaning. We need the intercept α to draw the line, but it has no statistical meaning.

12. Example 14.2 (continued…) The standard deviation σ, which describes the variability of the response y about the true regression line. The standard deviation σ, which describes the variability of the response y about the true regression line. The least-squares line estimates the true regression line. Recall that the residuals are the vertical deviations of the data points from the least-squares line: The least-squares line estimates the true regression line. Recall that the residuals are the vertical deviations of the data points from the least-squares line: Residual = observed y – predicted y = y - y ^

13. Standard Error About the Least-Squares Line We call this sample standard deviation a standard error to emphasize that it is estimated from data. We call this sample standard deviation a standard error 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)∑ residual 2 s = √ 1/(n – 2)∑ (y – y) 2 s = √ 1/(n – 2)∑ (y – y) 2 Use s to estimate the unknown σ in the regression model. Use s to estimate the unknown σ in the regression model. ^