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: Determine mutual - exclusivity

Mutually exclusive events cannot occur at the same time. Since the probability that a person is a liberal Democrat and believes the earth is warming is 0.18 (non - zero), they are not mutually exclusive.

Step2: Calculate \(P(\text{warming or liberal Democrat})\)

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Here, \(P(\text{warming}) = 0.6\), \(P(\text{liberal Democrat})=0.34\), and \(P(\text{warming and liberal Democrat}) = 0.18\). So \(P(\text{warming or liberal Democrat})=0.6 + 0.34-0.18=0.7600\).

Step3: Calculate \(P(\text{warming}|\text{liberal Democrat})\)

Use the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(P(A\cap B) = 0.18\) and \(P(B)=0.34\), so \(P(\text{warming}|\text{liberal Democrat})=\frac{0.18}{0.34}\approx0.5294\).

Step4: Calculate \(P(\text{warming}|\text{conservative Republican})\)

Use the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(P(A\cap B) = 0.11\) and \(P(B)=0.33\), so \(P(\text{warming}|\text{conservative Republican})=\frac{0.11}{0.33}\approx0.3333\).

Step5: Check for independence

Two events \(A\) (believing in warming) and \(B\) (party ideology) are independent if \(P(A|B)=P(A)\). Since \(P(\text{warming}|\text{liberal Democrat})\approx0.5294
eq P(\text{warming}) = 0.6\) and \(P(\text{warming}|\text{conservative Republican})\approx0.3333
eq P(\text{warming}) = 0.6\), belief in global warming and party ideology are dependent.

Step6: Calculate \(P(\text{mod/lib Republican}|\text{not warming})\)

Use the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(P(A\cap B) = 0.06\) and \(P(B)=0.34\), so \(P(\text{mod/lib Republican}|\text{not warming})=\frac{0.06}{0.34}\approx0.1765\).

Answer:

a. not mutually exclusive
b. 0.7600
c. 0.5294
d. 0.3333
e. belief in global warming and party ideology are dependent
f. 0.1765