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1、CHAPTER 11.1 (i) Ideally, we could randomly assign students to classes of different sizes. That is, each student is assigned a different class size without regard to any student characteristics such as ability and family background. For reasons we will see in Chapter 2, we would like substantial var
2、iation in class sizes (subject, of course, to ethical considerations and resource constraints).(ii) A negative correlation means that larger class size is associated with lower performance. We might find a negative correlation because larger class size actually hurts performance. However, with obser
3、vational data, there are other reasons we might find a negative relationship. For example, children from more affluent families might be more likely to attend schools with smaller class sizes, and affluent children generally score better on standardized tests. Another possibility is that, within a s
4、chool, a principal might assign the better students to smaller classes. Or, some parents might insist their children are in the smaller classes, and these same parents tend to be more involved in their childrens education.(iii) Given the potential for confounding factors some of which are listed in
5、(ii) finding a negative correlation would not be strong evidence that smaller class sizes actually lead to better performance. Some way of controlling for the confounding factors is needed, and this is the subject of multiple regression analysis.1.2 (i) Here is one way to pose the question: If two f
6、irms, say A and B, are identical in all respects except that firm A supplies job training one hour per worker more than firm B, by how much would firm As output differ from firm Bs?(ii) Firms are likely to choose job training depending on the characteristics of workers. Some observed characteristics
7、 are years of schooling, years in the workforce, and experience in a particular job. Firms might even discriminate based on age, gender, or race. Perhaps firms choose to offer training to more or less able workers, where “ability” might be difficult to quantify but where a manager has some idea abou
8、t the relative abilities of different employees. Moreover, different kinds of workers might be attracted to firms that offer more job training on average, and this might not be evident to employers.(iii) The amount of capital and technology available to workers would also affect output. So, two firm
9、s with exactly the same kinds of employees would generally have different outputs if they use different amounts of capital or technology. The quality of managers would also have an effect.(iv) No, unless the amount of training is randomly assigned. The many factors listed in parts (ii) and (iii) can
10、 contribute to finding a positive correlation between output and training even if job training does not improve worker productivity. 1.3 It does not make sense to pose the question in terms of causality. Economists would assume that students choose a mix of studying and working (and other activities
11、, such as attending class, leisure, and sleeping) based on rational behavior, such as maximizing utility subject to the constraint that there are only 168 hours in a week. We can then use statistical methods to measure the association between studying and working, including regression analysis that
12、we cover starting in Chapter 2. But we would not be claiming that one variable “causes” the other. They are both choice variables of the student. CHAPTER 22.1(i) Income, age, and family background (such as number of siblings) are just a few possibilities. It seems that each of these could be correla
13、ted with years of education. (Income and education are probably positively correlated; age and education may be negatively correlated because women in more recent cohorts have, on average, more education; and number of siblings and education are probably negatively correlated.)(ii) Not if the factor
14、s we listed in part (i) are correlated with educ. Because we would like to hold these factors fixed, they are part of the error term. But if u is correlated with educ then E(u|educ) 0, and so SLR.4 fails.2.2In the equation y = b0 + b1x + u, add and subtract a0 from the right hand side to get y= (a0
15、+ b0) + b1x + (u - a0). Call the new error e= u- a0, so that E(e)= 0. The new intercept is a0+ b0, but the slope is still b1.2.3(i) Let yi= GPAi, xi= ACTi, and n= 8. Then = 25.875, = 3.2125, (xi )(yi )= 5.8125, and (xi )2= 56.875. From equation (2.9), we obtain the slope as = 5.8125/56.875 .1022, ro
16、unded to four places after the decimal. From (2.17), = 3.2125 (.1022)25.875 .5681. So we can write = .5681 + .1022 ACT n = 8.The intercept does not have a useful interpretation because ACT is not close to zero for the population of interest. If ACT is 5 points higher, increases by .1022(5)= .511.(ii) The fitted values and residuals rounded to four decimal places are given along with
