Question

a 2010 pew research poll asked 1,306 americans \from what youve read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?\. the table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies.

	earth is warming	not warming	dont know (or refuse)	total
mod/lib republican	0.06	0.06	0.01	0.13
mod/cons democrat	0.25	0.07	0.02	0.34
liberal democrat	0.18	0.01	0.01	0.2
total	0.6	0.34	0.06	1

a) are believing that the earth is warming and being a liberal democrat mutually exclusive?
not mutually exclusive
mutually exclusive
b) what is the probability that a randomly chosen respondent believes the earth is warming or is a liberal democrat? (please round to four decimal places)
c) what is the probability that a randomly chosen respondent believes the earth is warming given that he is a liberal democrat? (please round to four decimal places)
d) what is the probability that a randomly chosen respondent believes the earth is warming given that he is a conservative republican? (please round to four decimal places)
e) does it appear that whether or not a respondent believes the earth is warming is independent of their party ideology?
belief in global warming and party ideology are dependent
belief in global warming and party ideology are independent
f) what is the probability that a randomly chosen respondent is a moderate/liberal republican given that he does not believe that the earth is warming? (please round to four decimal places)

Explanation:

Step1: Recall the definition of mutually - exclusive events

Two events \(A\) and \(B\) are mutually exclusive if \(P(A\cap B)=0\). Let \(A\) be the event that the earth is warming and \(B\) be the event of being a liberal Democrat. From the table, \(P(A\cap B) = 0.18
eq0\). So, they are not mutually exclusive.

Step2: Use the addition rule for probability

The addition rule is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Here, \(P(A) = 0.6\), \(P(B)=0.2\) and \(P(A\cap B)=0.18\). Then \(P(A\cup B)=0.6 + 0.2-0.18=0.62\).

Step3: Use the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). For the event \(A\) (earth is warming) and \(B\) (liberal Democrat), \(P(A|B)=\frac{0.18}{0.2}=0.9000\).

Step4: For conservative Republicans

Let \(A\) be the event that the earth is warming and \(B\) be the event of being a conservative Republican. \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), where \(P(A\cap B)=0.11\) and \(P(B)=0.33\). So \(P(A|B)=\frac{0.11}{0.33}\approx0.3333\).

Step5: Check for independence

Two events \(A\) (earth is warming) and \(B\) (party - ideology) are independent if \(P(A|B)=P(A)\). Since \(P(A)\) (probability that earth is warming) is \(0.6\), and \(P(A|B)\) values change depending on the party - ideology (e.g., \(P(A|B)\) for liberal Democrats is \(0.9\) and for conservative Republicans is \(0.3333\)), the belief in global warming and party ideology are dependent.

Step6: Use conditional probability formula

Let \(A\) be the event of being a moderate/liberal Republican and \(B\) be the event of not believing the earth is warming. \(P(A\cap B)=0.06\), \(P(B)=0.34\). Then \(P(A|B)=\frac{0.06}{0.34}\approx0.1765\).

Answer:

a) not mutually exclusive
b) 0.6200
c) 0.9000
d) 0.3333
e) belief in global warming and party ideology are dependent
f) 0.1765