A CBS News poll involved a nationwide random sample of 651 adults, asked those a

Question

A CBS News poll involved a nationwide random sample of 651 adults, asked those adults about their party affiliation (Democrat, Republican or none) and their opinion of how the US economy was changing ("getting better," "getting worse" or "about the same"). The results are shown in the table below.

If we randomly select one of the adults who participated in this study, compute: (round to four decimal places)

a.) P(worse|Democrat) =

b.) P(Democrat|worse) =

2.)

A 2010 Pew Research poll asked 1,306 Americans "From what you've 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.

a) Are believing that the earth is warming and being a liberal Democrat 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?

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)

3.)

A migraine is a particularly painful type of headache, which patients sometimes wish to treat with acupuncture. To determine whether acupuncture relieves migraine pain, researchers conducted a randomized controlled study where 89 females diagnosed with migraine headaches were randomly assigned to one of two groups: treatment or control. 43 patients in the treatment group received acupuncture that is specifically designed to treat migraines. 46 patients in the control group received placebo acupuncture (needle insertion at non-acupoint locations). 24 hours after patients received acupuncture, they were asked if they were pain free. Results are summarized in the contingency table below (Allais, 2011).
(round answers to 2 decimal places)

a) What percent of patients in the treatment group were pain free 24 hours after receiving acupuncture?  %
b) What percent of patients in the control group were pain free after 24 hours?  %
c) At first glance, does acupuncture appear to be an effective treatment for migraines? Explain your reasoning.

-no, because more than half of the individuals in the treatment group still experience pain 24 hours later

-yes, because a higher percentage of individuals in the treatment group were pain-free after 24 hours

-no, because there were fewer people in the treatment group

-yes, because more individuals in the treatment group were pain-free after 24 hours

d) Do the data provide convincing evidence that there is a real pain reduction for those patients in the treatment group? Or do you think that the observed difference might just be due to chance?

Yes, 23% is much higher than 4%

It is impossible to tell merely by comparing the sample proportions because the difference could be the result of random error in our sample

Explanation / Answer

Solution :

From given table,

better same worse Total

Republican 38 104 44 186

Democrat . 12 87 137 236

None 21 90 118 229

Total 71 281 299 651

(a) P(worse | Democrat) = 137 / 236 = 0.5806

(b)  P(Democrat | worse) = 137 / 299 = 0.4582

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