The American Community Survey is an ongoing survey that provides data every year to give communities the current information they need to plan investments and services. The 2010 American Community Survey estimates that 14.6% of Americans live below the poverty line, 20.7% speak a language other than English (foreign language) at home, and 4.2% fall into both categories. (a) Are living below the poverty line and speaking a foreign language at home disjoint?
No, there are people who are both living below the poverty line and speak a language other than English at home.
(b) Draw a Venn diagram summarizing the variables and their associated probabilities. The diagram should show two circles that overlap. The speak-FL circle has .165 in the non-overlap; the below PL circle has .104 in the non-overlap; the intersection has .042.
(c) What percent of Americans live below the poverty line and only speak English at home?
Each person living below the poverty line either speaks only English at home or doesnât. Since .146 of Americans live below the poverty line and .042 speak a language other than English at home, the other .104 only speak English at home.
(d) What percent of Americans live below the poverty line or speak a foreign language at home?
Using the General Addition Rule:
P(below P L or speak F L) = P(below P L) + P(speak F L) â P(both)
= 0.146 + 0.207 â 0.042 = 0.311
(e) What percent of Americans live above the poverty line and only speak English at home?
P(neither below PL nor speak FL) = 1 - P(below PL or speak FL) = 1 - 0.311 = 0.689
(f) Is the event that someone lives below the poverty line independent of the event that the person speaks a foreign language at home?
Two approaches: 1) Using the multiplication rule: P(below PL) * P(speak FL) = 0.146*0.207 = 0.030, which does not equal P(below PL and speak FL) = 0.042, therefore the events are dependent. 2) Using Bayesâ Theorem: If the two events are independent, then P(below PL
â speak FL) = P(below PL).
P(below P L | speak F L) = P(below P L and speak F L)/P(speak F L)
=0.042/0.207â 0.203