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Let's go through problem 4 step by step.

### Problem 4:
**Robbie is running for city mayor and conducted a survey of 120 people. Of those people, 47 said they would vote for Robbie.**

#### A) Construct a confidence interval with a margin of error of 3%.

To construct a confidence interval for the proportion, we will follow these steps:

1. **Determine the sample proportion (p-hat):**
   \[
   \hat{p} = \frac{\text{Number of people who said they would vote for Robbie}}{\text{Total number of people surveyed}} = \frac{47}{120} \approx 0.3917
   \]

2. **The margin of error is given as 3%:**
   \[
   \text{Margin of Error} (E) = 0.03
   \]

3. **Construct the confidence interval:**
   \[
   \text{Confidence Interval} = \hat{p} \pm E = 0.3917 \pm 0.03
   \]

   So, the confidence interval is:
   \[
   0.3917 - 0.03 \text{ to } 0.3917 + 0.03
   \]
   \[
   0.3617 \text{ to } 0.4217
   \]

   Converting to percentages:
   \[
   36.17\% \text{ to } 42.17\%
   \]

#### B) Do you think Robbie will win? Why or why not?

- Robbie's current support, based on the sample, is estimated to be between **36.17% and 42.17%**.
- To win, Robbie would typically need more than 50% of the vote.
- Since the confidence interval does not include 50%, this suggests that based on the current data, it is unlikely that Robbie will win unless something changes significantly in the electorate's opinion.

Would you like further details or explanations on this problem? Here are some related questions that might interest you:

1. How would the confidence interval change if the margin of error was smaller, say 2%?
2. What would the confidence interval be if the sample size was larger?
3. How does increasing the confidence level to 99% affect the confidence interval?
4. What are the implications of the confidence interval for campaign strategy?
5. How could Robbie improve his chances of winning based on statistical analysis?

**Tip:** Remember that confidence intervals give a range where the true population parameter is likely to fall, but they do not guarantee that the true parameter is within that range.

Explore how to construct a confidence interval with a margin of error of 3% in this statistics problem. Learn step-by-step how to estimate support for Robbie's mayoral campaign based on survey data. Understand the implications of the confidence interval for predicting electoral outcomes.

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