2.8 Poverty and language. 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.59 (a) Are living below the poverty line and speaking a foreign language at home disjoint

a) Draw a Venn diagram summarizing the variables and their associated probabilities

b) What percent of Americans live below the poverty line and only speak English at home?

c) What percent of Americans live below the poverty line or speak a foreign language at home?

a) No, living below the poverty line and speaking a foreign language at home are not disjoint.

A) The Venn diagram is presented in the attached image to this solution

B) percent of Americans living below the poverty line and only speak English at home = 10.4%

C) The percent of Americans living below the poverty line or speak a foreign language at home = 31.1%

Step-by-step explanation:

Let the set of people living below the poverty line be represented by L

And the set of people speaking a foreign language at home be represented by S

P(L) = 14.6% = 0.146

P(S) = 20.7% = 0.207

P(L n S) = 4.2% = 0.042

The set of complements, that is, set of people not living below the poverty line is L'. P(L') = 100 - 14.6 = 85.4% = 0.854

And the set of people that speak English at home is S'. P(S') = 100 - 20.7 = 79.3% = 0.793

a) No, living below the poverty line and speaking a foreign language at home are not disjoint.

This is because the living below the poverty line and speaking a foreign language are events that have elements in common, that is, one variable falls in the other one. they are not disjoint events.

A) The Venn diagram is presented in the attached image to this solution

B) The percent of Americans living below the poverty line and only speak English at home = P(L n S') = P(L) - P(L n S) = 0.146 - 0.042 = 0.104 = 10.4%

C) percent of Americans living below the poverty line or speak a foreign language at home P(L u S) = P(L n S') + P(L' n S) + P(L n S)

P(L n S') = P(L) - P(L n S) = 0.104

P(L' n S) = P(S) - P(L n S) = 0.207 - 0.042 = 0.165

P(L n S) = 0.104

P(L u S) = P(L n S') + P(L' n S) + P(L n S) = 0.104 + 0.165 + 0.042 = 0.311 = 31.1%

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

probability for j is 1/4

while

a is 1/6