1. AP Statistics Section 15 A

2. The Regression Model When a scatterplot shows a linear relationship between a quantitative explanatory variable x and a quantitative response variable y, we can use the ____________ line fitted to the data to predict y for a given x value. Now we want to do tests and confidence intervals in this setting.

3. Example 15.1: Infants who cry easily may be more easily stimulated than others. This may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants four to ten days old and their later IQ test scores. A snap of a rubber band on the sole of the foot caused the infants to cry. The researchers recorded the crying and measured its intensity by the number of peaks in the most active 20 seconds. They later measured the children’s IQ at age three years using the Stanford-Binet IQ test. The table at the right contains data on 38 infants.

4. CryingIQCryingIQCryingIQCryingIQ 1087209017941294 1297161001910312103 9 231031310414106 16106271081810910109 18109151121811223113 1511421114161189119 12119121201912016124 20132151332213531135 16136171413015522157 3315913162

5. Let’s first of all look at the scatterplot of the data. Describe what you see (strength, direction, shape and unusual points).

6. Because the scatterplot show a roughly linear pattern, the correlation, r, describes The correlation between crying and IQ is r = _____.

7. We are interested in predicting the response variable from information given about the explanatory variable. We find the least squares regression line (LSR) for predicting IQ from crying. The equation for the least squares regression line is

8. Interpret the slope of this LSR line. Interpret the y-intercept for this LSR line.

9. The coefficient of determination, r 2, for this data is ______. Interpret this value.

10. Calculate the residual for an infant who has 33 crying peaks.

11. Conditions for the Regression Model Because we calculate them from the sample data, the slope b and the intercept a of the LSL are statistics. These statistics would take somewhat different forms if we repeated the study with different infants. To do formal inference, we need to think of a and b as estimates of population parameters.

12. The required conditions for regression inference are: * The data comes from an SRS of the population. * The observations are independent. In particular, repeated observations of the same individual are not allowed. So we can’t make inferences about the growth of a single child over time. * The true relationship is linear. Look at the scatterplot to check that the overall pattern is roughly linear. A plot of residuals against x magnifies any unusual pattern. What do we look for?

14. * The standard deviation of y is the same for all values of x. The scatter of the residuals above and below the line y = 0 should be roughly the same over the entire range of the data. It is quite common to find that as the response y gets larger, so does the scatter of the points about the fitted line (i.e. a fan shape). This means that the standard deviation is changing. You cannot safely use our inference procedures when this happens.

15. * For any fixed value of x, the y-values vary according to a Normal distribution. Because we rarely have enough observations at each x- value, we make one graph of all the residuals and check for clear skewness or other major departures from Normality.