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Find the joint and marginal relative frequencies. Then use these values to find the conditional probability.

[tex]\[
P(\text{yes} \mid \text{east side}) = \frac{P(\text{east side and yes})}{P(\text{east side})} = \frac{0.47}{0.47 + 0.08} \approx 0.855
\][/tex]

So, the probability that a resident of the east side of the city will support the project is about 85.5%.

1. What is the probability that a randomly selected resident who does not support the project in the example above is from the west side?
2. After a conference, 220 men and 270 women respond to a survey. Of those, 200 men and 230 women say the conference was impactful. Organize these results in a two-way table. Then find and interpret the marginal frequencies.



Answer :

GinnyAnswer
Sure, let's systematically solve the problem of organizing the survey results into a two-way table and finding and interpreting the marginal frequencies.

### Step-by-Step Solution

Step 1: Organizing the Survey Results

The survey results are as follows:
- Total men surveyed: 220
- Men who found the conference impactful: 200
- Men who did not find the conference impactful: [tex]\(220 - 200 = 20\)[/tex]

- Total women surveyed: 270
- Women who found the conference impactful: 230
- Women who did not find the conference impactful: [tex]\(270 - 230 = 40\)[/tex]

- Total participants surveyed: [tex]\(220 + 270 = 490\)[/tex]
- Total participants who found the conference impactful: [tex]\(200 + 230 = 430\)[/tex]
- Total participants who did not find the conference impactful: [tex]\(20 + 40 = 60\)[/tex]

We can organize these results into a two-way table:

| | Impactful | Not Impactful | Total |
|------------------|-----------|---------------|-------|
| Men | 200 | 20 | 220 |
| Women | 230 | 40 | 270 |
| Total | 430 | 60 | 490 |

Step 2: Calculating and Interpreting Marginal Frequencies

Marginal frequencies represent the totals for each row and column. They help us understand the distribution across different categories. From the two-way table, we can determine the marginal frequencies:

1. Impactful:
- Total participants who found the conference impactful: 430

2. Not Impactful:
- Total participants who did not find the conference impactful: 60

3. Total:
- Total participants surveyed: 490

To summarize, the marginal frequencies are:
- 430 participants found the conference impactful.
- 60 participants did not find the conference impactful.
- The total number of participants surveyed was 490.

#### Interpretation:
- Impactful (430) indicates that most survey participants believed the conference had an impact.
- Not Impactful (60) shows a smaller group of participants felt the conference was not impactful.
- Total (490) confirms the overall number of respondents, derived from adding both men and women's responses.

This detailed interpretation illustrates how the participants' opinions are distributed, which can be valuable for understanding the conference's perceived effectiveness among different demographics.

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