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The Regression ModelWhen 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.
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.
Use the Data Analysis Toolbox to analyze these data.Who? We are told only that the individuals in the studyWhat? 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.
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, describesThe correlation between crying and IQ is r = _____.
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 : ______________
The coefficient of determination, r2, for this data is ______. Interpret this value.
Calculate the residual for an infant who has 10 crying peaks. The LSL is
Conditions for the Regression ModelBecause 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?
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.
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.
