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.

	better	same	worse
democrat	12	87	137
none	21	90	118

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

a. p(affiliated with neither party) =

b. p(worse) =

c. p(worse|affiliated with neither party) =

d. p(affiliated with neither party|worse) =

e. p(affiliated with neither party and worse) =

Explanation:

Step1: Calculate total number of participants

The total number of adults in the sample is \(38 + 104+44 + 12+87+137+21+90+118=651\).

Step2: Calculate P(affiliated with neither party)

The number of adults affiliated with neither party is \(21 + 90+118 = 229\). So \(P(\text{affiliated with neither party})=\frac{229}{651}\approx0.3518\).

Step3: Calculate P(worse)

The number of adults who think the economy is getting worse is \(44 + 137+118=299\). So \(P(\text{worse})=\frac{299}{651}\approx0.4593\).

Step4: Calculate P(worse|affiliated with neither party)

By the formula for conditional - probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here \(A\) is the event that the economy is getting worse and \(B\) is the event of being affiliated with neither party. \(P(A\cap B)=\frac{118}{651}\) and \(P(B)=\frac{229}{651}\), so \(P(\text{worse}|\text{affiliated with neither party})=\frac{118}{229}\approx0.5153\).

Step5: Calculate P(affiliated with neither party|worse)

By the formula for conditional - probability \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). Here \(A\) is the event that the economy is getting worse and \(B\) is the event of being affiliated with neither party. \(P(A\cap B)=\frac{118}{651}\) and \(P(A)=\frac{299}{651}\), so \(P(\text{affiliated with neither party}|\text{worse})=\frac{118}{299}\approx0.3947\).

Step6: Calculate P(affiliated with neither party and worse)

The number of adults who are affiliated with neither party and think the economy is getting worse is \(118\). So \(P(\text{affiliated with neither party and worse})=\frac{118}{651}\approx0.1813\).

Answer:

a. \(0.3518\)
b. \(0.4593\)
c. \(0.5153\)
d. \(0.3947\)
e. \(0.1813\)