In a survey of 

22642264 

​adults, 

729729 

say they believe in UFOs. 

Construct a 

99 %99% 

confidence interval for the population proportion of adults who believe in UFOs.




To construct a **99% confidence interval** for the population proportion of adults who believe in UFOs, follow these steps.

---

### Step 1: Define Variables
- **Sample size \(n\)** = 22,642,264
- **Number of successes \(x\)** = 729,729 (adults who believe in UFOs)
- **Sample proportion \( \hat{p} \)** = \( \frac{x}{n} \)  
  \[
  \hat{p} = \frac{729,729}{22,642,264} \approx 0.0322
  \]

### Step 2: Find the Critical Value \( Z_{\alpha/2} \)
For a **99% confidence interval**, the critical value \( Z_{\alpha/2} \) is approximately 2.576 (from the standard normal distribution).

### Step 3: Calculate the Standard Error
The standard error \( SE \) for a population proportion is:

\[
SE = \sqrt{ \frac{\hat{p}(1 - \hat{p})}{n} }
\]

\[
SE = \sqrt{ \frac{0.0322 \times (1 - 0.0322)}{22,642,264} } \approx 0.000036
\]

### Step 4: Compute the Confidence Interval
The formula for the confidence interval is:

\[
\hat{p} \pm Z_{\alpha/2} \times SE
\]

\[
0.0322 \pm 2.576 \times 0.000036
\]

\[
0.0322 \pm 0.0000927
\]

### Step 5: Final Interval
\[
(0.0321, 0.0323)
\]

Thus, the **99% confidence interval** for the proportion of adults who believe in UFOs is approximately:

\[
(0.0321, 0.0323)
\]

---

Would you like more details or have any questions about these calculations?  

---

### 5 Related Questions  
1. How does the choice of confidence level affect the width of a confidence interval?  
2. What would the confidence interval look like if the sample size was much smaller?  
3. How can we interpret this confidence interval in real-world terms?  
4. What is the margin of error, and how is it connected to the confidence interval?  
5. Why do we assume a normal distribution when calculating confidence intervals for proportions?

---

**Tip:** Larger sample sizes result in narrower confidence intervals because the standard error becomes smaller.

In a survey of 22,642,264 adults, 729,729 say they believe in UFOs. Construct a 99% confidence interval for the population proportion of adults who believe in UFOs.

Construct a 99% confidence interval for the proportion of adults who believe in UFOs. Based on a survey of 22,642,264 adults with 729,729 believing in UFOs, we use statistical formulas to calculate the interval, applying concepts such as sample proportion, standard error, and the central limit theorem.

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