a cbs news poll involved a nationwide random sample of 651 adults, asked those adults about their party…

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. \n| | better | same | worse |\n|--|--|--|--|\n| republican | 38 | 104 | 44 |\n| democrat | 12 | 87 | 137 |\n| none | 21 | 90 | 118 |\nif we randomly select one of the adults who participated in this study, compute: (round to four decimal places)\na. p(republican) =\nb. p(worse) =\nc. p(worse|republican) =\nd. p(republican|worse) =\ne. p(republican and worse) =

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. \n| | better | same | worse |\n|--|--|--|--|\n| republican | 38 | 104 | 44 |\n| democrat | 12 | 87 | 137 |\n| none | 21 | 90 | 118 |\nif we randomly select one of the adults who participated in this study, compute: (round to four decimal places)\na. p(republican) =\nb. p(worse) =\nc. p(worse|republican) =\nd. p(republican|worse) =\ne. p(republican and worse) =

Answer

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(Republican)

The number of Republicans is $38 + 104+44 = 186$. So $P(\text{Republican})=\frac{186}{651}\approx0.2857$.

Step3: Calculate P(worse)

The number of people who think the economy is getting worse is $44 + 137+118 = 300$. So $P(\text{worse})=\frac{300}{651}\approx0.4608$.

Step4: Calculate P(worse|Republican)

Using 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 a Republican. $P(\text{worse}\cap\text{Republican})=\frac{44}{651}$ and $P(\text{Republican})=\frac{186}{651}$. So $P(\text{worse}|\text{Republican})=\frac{44}{186}\approx0.2366$.

Step5: Calculate P(Republican|worse)

$P(\text{Republican}|\text{worse})=\frac{P(\text{Republican}\cap\text{worse})}{P(\text{worse})}$. Since $P(\text{Republican}\cap\text{worse})=\frac{44}{651}$ and $P(\text{worse})=\frac{300}{651}$, then $P(\text{Republican}|\text{worse})=\frac{44}{300}\approx0.1467$.

Step6: Calculate P(Republican and worse)

The number of Republicans who think the economy is getting worse is 44. So $P(\text{Republican and worse})=\frac{44}{651}\approx0.0676$.

Answer:

a. $0.2857$ b. $0.4608$ c. $0.2366$ d. $0.1467$ e. $0.0676$