Question

Math

Posted 4 months ago

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Leon is interested in the relationships among geographic region and political affiliation. Leon collected data about the political parties of US senators at that time and the regions of the US that they represent. His results are in the table below.
\begin{tabular}{lllll}
Region & Democrats & Republicans & Independents & Total \\
\hline Southeast & 6 & 20 & 0 & 26 \\
Southwest & 1 & 5 & 0 & 6 \\
West & 8 & 10 & 0 & 18 \\
Midwest & 13 & 11 & 0 & 24 \\
Northeast & 14 & 7 & 1 & 22 \\
Pacific & 2 & 2 & 0 & 4 \\
Total & 44 & 55 & 1 & 100
\end{tabular}
Which of the following interpretations are correct?
Choose all answers that apply:
A Given that a senator was in the Republican Party, the senator was more likely to represent the midwest than any other region.
B Given that a senator is an Independent, it is certain that the senator represented the northeast.
C Given that a senator represents the southeast, it is likely that the senator is in the Republican Party.
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

According to the table, there are 11 Republican senators in the Midwest

step 3

The region with the highest number of Republican senators is the Southeast, with 20 Republican senators

step 4

Since 20 (Southeast) is greater than 11 (Midwest), statement A is incorrect

step 5

To verify statement B, we look at the number of Independent senators in each region

step 6

There is only 1 Independent senator, and they represent the Northeast

step 7

Therefore, statement B is correct because it is certain that the Independent senator represented the Northeast

step 8

To assess statement C, we examine the likelihood of a senator from the Southeast being a Republican

step 9

There are 26 senators in total from the Southeast, with 20 being Republicans

step 10

The probability that a senator from the Southeast is a Republican is $\frac{20}{26}$, which is approximately 0.77

step 11

Since this probability is high, statement C is correct

1 Answer

B, C

Key Concept

Conditional Probability and Certain Events

Explanation

To determine the correctness of the statements, we used the concepts of conditional probability for statements A and C, and the concept of certainty for statement B. Conditional probability assesses the likelihood of an event given that another event has occurred, while certainty implies that an event will definitely occur.

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