AP Statistics Section 15 A

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.

squaresleast

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.

Crying IQ Crying IQ Crying IQ Crying IQ

10 87 20 90 17 94 12 94

12 97 16 100 19 103 12 103

9 103 23 103 13 104 14 106

16 106 27 108 18 109 10 109

18 109 15 112 18 112 23 113

15 114 21 114 16 118 9 119

12 119 12 120 19 120 16 124

20 132 15 133 22 135 31 135

16 136 17 141 30 155 22 157

33 159 13 162

Use the Data Analysis Toolbox to analyze these data. Who? We are told only that the individuals in the study

What? The explanatory variable is _____________ and the response variable is_________.

Why? Researchers wanted to determine if

When, where, how and by whom? The data come from an experiment described in 1964 in the journal Child Development.

old. years 3 again when

and old days 10 - 4 when studied infants 38 are

intensity cryingscore IQ

3 ageat IQ and infants as

intensity crying ebetween thn associatioan is there

Examine the data on a scatterplot of the paired data. Look for form, direction and strength of the relationship as

well as outliers and other deviations. There is a (weak/moderate/ strong) (negative/no/positive)

(linear/nonlinear)relationship (with/with no) extreme outliers. There (are/are no) potentially influential

observations.

Because the scatterplot show a roughly linear pattern, the correlation, r, describes

The correlation between crying and IQ is r = _____.

ip.relationsh theofstrength anddirection the

.455

We are interested in predicting the response from information given about the explanatory variable. We find the least squares regression

line for predicting IQ from crying. The equation for the least squares regression line is :

____________________268.91493.1ˆ xy

Interpret the slope of this LSR line.

268.91493.1ˆ xy

points 1.493ely approximatby increase will

score IQ theintensity, crying in the 1 of increaseevery For

The coefficient of determination, r2, for this data is ______. Interpret this value.207.

IQ. andintensity

cryingbetween iprelationshlinear by the explained

becan score IQ in the variation theof 20.7%About

Calculate the residual for an infant who has 10 crying peaks. The LSL is

268.91493.1ˆ xy

valuey - valueobserved residual

-19.2106.2 - 87

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. The required

conditions for regression inference are:The observations are ______________. 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 _______. 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?

tindependen

linear

residuals in thepattern a

The standard deviation of the response about the line is __________ everywhere. The scatter of the data 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. This means that the standard deviation, , is changing. You cannot safely use our inference procedures when this happens.

same the

The response varies _________ about the true regression line. Make a histogram or stemplot of the residuals and check for clear skewness or other major

departures from Normality.

Normally