AP STATISTICS LESSON 14 – 1 ( DAY 1 )

1. AP STATISTICSLESSON 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 standard errors for regression lines. • 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.

4. Example 14.1 Page 781Crying and IQ • Plot and interpret. • Numerical summary • 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. 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.

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. • 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 μy has a straight-line relationship with x: μy = 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.

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 μy = α +βx says that the mean response μy 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

10. Inference 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.

11. Example 14.2 Page 784Slope and Intercept 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. 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 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. The standard error about the line is s = √ 1/(n – 2)∑ residual2 s = √ 1/(n – 2)∑ (y – y)2 Use s to estimate the unknown σ in the regression model. ^