Student achievement and self-concept. In order to determine if student achievement is related to self-concept, as measured by the Piers-Harris Children's Self-Concept Scale, data were collected on 78 seventhgrade students from a rural midwestern school. Table 28.14 shows the records for the first 10 students on the following variables:16
We will investigate the relationship between GPA and only three of the explanatory variables:
¦ IQ, the student's score on a standard IQ test
¦ C2, the student's self-assessment of his or her school status
¦ C5, the student's self-assessment of his or her popularity
Use statistical software to analyze the relationship between students' GPA and their IQ, self-assessed school status (C2), and self-assessed popularity (C5).
(a) One observation is an extreme outlier when all three explanatory variables are used. Which observation is this? Give the observation number and explain how you found it using regression output. Find this observation in the data list. What is unusual about it?
(b) Software packages often identify unusual or influential observations. Have any observations been identified as unusual or influential? If so, identify these points on a scatterplot of GPA versus IQ.
(c) C2 (school status) is the aspect of self-concept most highly correlated to GPA. If we carried out the simple linear regression of GPA on C2, what percent of the variation in students' GPAs would be explained by the straight-line relationship between GPA and C2?
(d) You know that IQ is associated with GPA, and you are not studying that relationship. Because C2 and IQ are positively correlated (r =0.547), a significant relationship between C2 and GPA might occur just because C2 can "stand in" for IQ. Does C2 still contribute significantly to explaining GPA after we have allowed for the relationship between GPA and IQ? (Give a test statistic, its P-value, and your conclusion.)
(e) A new student in this class has IQ 115 and C2 score 14. What do you predict this student's GPA to be? (Just give a point prediction, not an interval.)