Question

Here is a variable: Some people support a government-backed health insurance plan that would cover all citizens, whereas others oppose such a plan. This variable has two values: “support government plan” and “oppose government plan.” Let's say that one explanation, the “income explanation,” suggests this hypothesis: In a comparison of individuals, those who have low incomes will be more likely to support a government plan than will those who have high incomes. A rival explanation, the “age explanation,” suggests this hypothesis: In a comparison of individuals, those who are older will be more likely to support a government health plan than will those who are younger. For the purposes of this exercise, income will be X, age will be Z, and support for government-backed health insurance will be Y. A. Suppose that after controlling for age (Z) , the relationship between income (X) and health insurance opinions (Y) turned out to be spurious. (i) Using this chapter's discussion of spuriousness as a guide, write 4-5 sentences explaining how the Z-Y relationship and the Z-X relationship would produce a spurious relationship between X and Y. (ii) Sketch a line graph depicting a spurious relationship between X and Y, controlling for Z. The vertical axis will show the percentage supporting a government plan. Invent plausible percentages for the values of Y. The horizontal axis will show the two values of income. There will be two lines inside the graph: one for younger people and one for older people. B. Suppose that after controlling for age (Z), the relationship between X, Z, and Y turned out to be additive. (i) Using this chapter's discussion of additive relationships as a guide, write 4-5 sentences explaining how this set of relationships would fit an additive pattern. (ii) Sketch a line graph depicting the additive relationships between X, Z, and Y. The vertical axis will show the percentage supporting a government plan. Just as you did in part A, invent plausible percentages for the values of Y. The horizontal axis will show the two values of income. There will be two lines inside the graph: one for younger people and one for older people. C. Suppose that a set of interaction relationships exists between X, Y, and Z. Suppose further that the interaction takes this form. Among younger people, income has no effect on the dependent variable; among older people, those with lower incomes are more likely to support a government plan that are those with high incomes. Sketch a line graph depicting this set of interaction relationships. Remember that the vertical axis shows the percentage supporting a government plan. As before, invent plausible percentages for the values of Y. The horizontal axis will show the two values of income. There will be two lines inside the graph: one for younger people and one for older people.

Answer 1

Note:

Hey there! Thank you for the question. As questions (A) and (B) are two different questions, according to our policy, we have solved only question (A) (both parts) for you. Since the discussion in the chapter on spuriousness is not available to us, we have answered the question to the best of our ability.

(A) (i)

Suppose two variables reveal at least a moderate level of correlation among them. However, when trying to understand the situation logically, it seems very unlikely that there can be any true relationship between the variables. In such a situation, most of the times, a deeper investigation reveals that there is a third variable that is strongly (and logically) related to each of these variables, separately, and the relationship of the two variables with the third variable leads to a situation where it appears that the two unrelated variables are also seemingly related. This is spurious correlation.

In this case, it is logical that income (X) would be strongly related to age (Z), as very young and very old people (who either have very less experience or a declining efficiency level) will have low income, whereas the people in the middle age group would have comparatively higher income (due to an adequate amount of experience, as well as a steady efficiency).

Again, it is logical that age (Z) would affect health insurance opinions (Y), as people at an older age (and possible retired) often face declining health conditions, and due to the high medical treatment costs, would naturally be glad to have government support. On the other hand, younger people might not feel the need of government support, as they naturally have more health and vigour, as well as the means to pay for health-care facilities.

When the difference in health insurance opinions and incomes are not controlled by age, it might appear that income (X) and health insurance opinions (Y) are strongly correlated. However, if age (Z) is controlled for, then, for people in a particular age group, the income (X) and health insurance opinions (Y) would vanish, and no strong correlation would be observed. This is spurious correlation in this context.

(A) (ii)

We made up the following data:

X ($ per year)	Y (percentage supporting government plan)	Z
100000	20	Young
500000	55	Old
100000	31	Young
500000	51	Old
100000	22	Young
500000	63	Old

The line graph is shown below: