A random sample of 38 known exoplanets is selected, and 17 are found to exist in multiple-star systems.

Compute a 95% confidence interval for the true proportion of multi-star exoplanets among all detected exoplanets. Give your final answer with at least 3
dp.

Lower bound of the 95% CI (SMALLER NUMBER HERE):	Answer 1 Question 1
Upper bound of the 95% CI (BIGGER NUMBER HERE):	Answer 2 Question 1

To calculate the 95% confidence interval (CI) for the true proportion of exoplanets that exist in multiple-star systems, we will use the following formula for a confidence interval for a population proportion:

\[
\text{CI} = \hat{p} \pm Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\]

Where:
- \(\hat{p}\) is the sample proportion.
- \(Z_{\alpha/2}\) is the z-score corresponding to the desired confidence level.
- \(n\) is the sample size.

### Step 1: Calculate the sample proportion \(\hat{p}\)
\[
\hat{p} = \frac{17}{38}
\]

### Step 2: Determine the z-score for a 95% confidence level
For a 95% confidence interval, the z-score (\(Z_{\alpha/2}\)) is 1.96.

### Step 3: Plug the values into the formula and calculate the confidence interval

Let's compute the values.The 95% confidence interval for the true proportion of multi-star exoplanets among all detected exoplanets is:

- **Lower bound** (smaller number): 0.289
- **Upper bound** (bigger number): 0.605

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. What factors could influence the proportion of exoplanets found in multi-star systems?
2. How would the confidence interval change with a larger sample size?
3. What is the interpretation of this confidence interval in the context of exoplanet research?
4. How does the z-score affect the width of the confidence interval?
5. What are some other statistical methods used in analyzing exoplanet data?

**Tip:** Increasing the sample size reduces the margin of error, leading to a narrower confidence interval.

Learn how to calculate a 95% confidence interval for the true proportion of multi-star exoplanets among all detected exoplanets. Includes step-by-step calculation with specific data points: sample size, sample proportion, and z-score. Suitable for advanced learners and researchers in astronomy and statistics.

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