Homework, ECO220Y: Lectures 7 & 8 PROBLEM SOLUTIONS For Part e. of Exercise 4.58 the solution is not given in the back of your textbook. Here is my answer: There is a strong positive linear association between study time and marks. The coefficient of correlation is 0.8811, which is close to its maximum possible value of one for a perfect positive linear association. The coefficient of determination is 0.7763, which means that about 78 percent of the variation among student marks moves with variation in study time. In addition to the coefficients of correlation and determination we may look at the least squares line: marks-hat = 5.917 + 1.705*study-time. In these data we see that students who spend an extra hour of studying have course grades that are on average 1.7 points higher. The intercept of 5.917 has no interpretation because none of our sample has reported anything near zero hours of study time. It is simply a shifter of the least squares line. Saying something like Studying zero hours means a grade of 5.9 is incorrect for two reasons: (1) We have absolutely no data to support that estimate of what marks would be if a student studied zero hours and (2) It wrongly implies causality, which is discussed next. We cannot conclude that the increased study time is causing the higher marks. In other words, we cannot say things like if a student studies an extra hour then their course grade would go up by 1.7 points or each extra hour of study will increase the course grade by 1.7 points. Statements like these are wrong because they imply that we have estimated the causal effect, which we have not. What we can do is describe the data. It is true that in the data students that studied more hours have higher marks and that in the data we see that students that studied an extra hour had marks that are 1.7 points higher. These are correct descriptive statements and are different from causal statements. (In this course we will return to this issue again and go even deeper.) (1) No. There are many problems with that conclusion. First, covariance only measures LINEAR relationships. It does NOT measure non-linear relationships. So we could not conclude anything about the strength of a relationship only about the strength of a linear relationship. Second, covariance is sensitive to the units of measurement (units of the covariance are units of the first variable times units of the second variable). Because you have been given no information about what X, Y, W, and Z measure and in what units they are measured the given covariance numbers are meaningless. We do not know that they are measured in comparable units. Third, the variability of each of the variables affects the magnitude of the covariance independent of any relationship between the variables. We have not been given any information about sX, sY, sW, and sZ. Fourth, even if we were talking only about linear relationships, the units of measurement were the same, and the variance of the variables were comparable, -20 would indicate a stronger relationship than 10. (2) To answer this question we need to use the least squares method. However, the information needed has not been directly given: we must figure out the covariance.
r=
s XY s X sY s XY 10 *10 = 50
slope(y / x) = s XY 50 = = 0.5 2 s X 10 2
0.5 = s XY
Since $10 to $12 is a $2 increase in x, we would expect it to be associated with a $2*0.5 = $1 increase in y. (3)
r=
s XY 12 = = 1.5 s X sY 2 * 4
BUT we know that the coefficient of correlation (r) must always be between -1 and 1: -1 r 1. Hence something is wrong: in particular, it is mathematically impossible to have such a negative covariance with such small standard deviations. Yes, (4) they can be NON-linearly related. For example, a symmetric parabola has a relationship, but if fit a line it will be horizontal.
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(5) (a) These data are cross-sectional. Both variables contain interval data. (We will also learn in future lectures that these data are observational.) (b) These results show that there is a negative association between the number of children women have and years of education. In particular, each extra year of education is associated with 0.2 fewer children on average. This is NOT to say that we can conclude that education is causing women to have fewer children. What we have observed is simply a CORRELATION between these variables. Correlations do not imply causal relationships. All we can do is describe what we see in the data, which is simply that the average number of children more educated women have is lower than the average number of children less educated women have. We cannot conclude that the education itself is causing the measured difference (as per the slope of the OLS line). (To convince yourself, think of reasons that women choose to obtain more education and ask if any of those underlying factors would also affect their choice about having children.) The coefficient of correlation (-0.3132) shows that there is a relatively weak negative relationship between these two variables. The intercept of 5.1 is meaningless. No Canadian women in our sample had zero years of education (not even close to zero years of education). We should not extrapolate back because it would take us well out of our sample and hence wed have NO evidence to support an interpretation of 5.1 as the number of children women with zero years of education would have on average. Furthermore, why would we even care about the mathematical intercept in this particular example? Do you think there are many (any?) women in the Canadian population that have zero years of education? If not, what would we be making an inference about? (c) The scatter diagram is not a good summary of these interval variables because in this example both variables are integers and take on relatively few unique values. You should add more explanation. An alternate way to summarize these...