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	0.18	0.07	0.02	0.27
liberal democrat	0.25	0.01	0.01	0.27
total	0.6	0.34	0.06	1

a) are believing that the earth is warming and being a liberal democrat mutually exclusive?

• mutually exclusive
• not 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 independent
• belief in global warming and party ideology are dependent

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: Check mutual - exclusivity

Mutually exclusive events cannot occur at the same time. Since 0.25 of liberal Democrats believe the earth is warming, they can be both a liberal Democrat and believe in earth - warming. So they are not mutually exclusive.

Step2: Probability of believing earth is warming and being a Democrat

Sum the probabilities of Mod/Cons Democrat and Liberal Democrat who believe the earth is warming: \(0.06 + 0.25=0.31\)

Step3: Probability of believing earth is warming or being a liberal Democrat

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be the event of believing the earth is warming (\(P(A) = 0.6\)) and \(B\) be the event of being a liberal Democrat (\(P(B)=0.34\)), and \(P(A\cap B) = 0.25\). Then \(P(A\cup B)=0.6 + 0.34-0.25 = 0.69\)

Step4: Probability of believing earth is warming given being a liberal Democrat

Use the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here \(A\) is believing the earth is warming and \(B\) is being a liberal Democrat. \(P(A\cap B) = 0.25\) and \(P(B)=0.34\), so \(P(A|B)=\frac{0.25}{0.34}\approx0.7353\)

Step5: Check independence

If \(P(A|B)=P(A)\), then \(A\) and \(B\) are independent. \(P(A) = 0.6\) and \(P(A|B)\) (e.g., for liberal Democrats) is \(\frac{0.25}{0.34}\approx0.7353
eq0.6\). So belief in global warming and party ideology are dependent.

Step6: Probability of not believing earth is warming given being a Mod/Lib Republican

Use the formula for conditional probability. The probability of being a Mod/Lib Republican and not believing the earth is warming is \(0.2\), and the probability of being a Mod/Lib Republican is \(0.11 + 0.2+0.02=0.33\). So \(P=\frac{0.2}{0.33}\approx0.6061\)

Answer:

a) not mutually exclusive
b) \(0.31\)
c) \(0.69\)
d) \(0.7353\)
e) belief in global warming and party ideology are dependent
f) \(0.6061\)