A survey of 2302 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 406 have donated blood in the past two years. Complete parts​ (a) through​ (c) below.
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Part 1
​(a) Obtain a point estimate for the population proportion of adults in the country aged 18 and older who have donated blood in the past two years.
ModifyingAbove p with caretequals
  
0.176
​(Round to three decimal places as​ needed.)
Part 2
​(b) Verify that the requirements for constructing a confidence interval about p are satisfied.
The sample 
can be assumed to be
 a simple random​ sample, the value of 
n ModifyingAbove p with caret left parenthesis 1 minus ModifyingAbove p with caret right parenthesis
 is 
  
enter your response here​, which is 
greater than or equal to
 ​10, and the 
sample size
 
can be assumed to be
 less than or equal to​ 5% of the 
population size.
​(Round to three decimal places as​ needed.)

Let's tackle the problem step by step.

### Part 1: Point Estimate for the Population Proportion

The point estimate for the population proportion \( p \) is calculated using the formula:

\[
\hat{p} = \frac{x}{n}
\]

Where:
- \( x \) = number of adults who donated blood in the past two years (406),
- \( n \) = total number of adults surveyed (2302).

\[
\hat{p} = \frac{406}{2302} \approx 0.176
\]

Thus, the point estimate \( \hat{p} = 0.176 \) is already given.

---

### Part 2: Verify the Requirements for Constructing a Confidence Interval

To verify the requirements for constructing a confidence interval, we need to check two key conditions:

1. **The Sample Size Condition:**
   The sample must be large enough. This is typically verified using the expression:

   \[
   n \hat{p}(1 - \hat{p}) \geq 10
   \]

   Let's calculate it:
   - \( \hat{p} = 0.176 \),
   - \( n = 2302 \).

   \[
   n \hat{p}(1 - \hat{p}) = 2302 \times 0.176 \times (1 - 0.176)
   \]

   First, calculate \( 1 - \hat{p} = 1 - 0.176 = 0.824 \).

   Now compute:

   \[
   n \hat{p}(1 - \hat{p}) = 2302 \times 0.176 \times 0.824 \approx 334.24
   \]

   Since \( 334.24 \geq 10 \), this condition is satisfied.

2. **The Independence Condition:**
   The sample size must be less than or equal to 5% of the population. Since the survey sample includes 2302 people, we need to check that:

   \[
   n \leq 0.05 \times \text{population size}
   \]

   As long as the population is large enough (greater than 46,040 people), this condition is satisfied. In a large country, it is reasonable to assume this holds.

---

### Summary:

- The point estimate for the population proportion is \( \hat{p} = 0.176 \).
- The conditions for constructing a confidence interval are met because:
  - \( n \hat{p}(1 - \hat{p}) = 334.24 \geq 10 \),
  - The sample size of 2302 is likely less than 5% of the population.

Do you have any questions, or would you like further details? Here are some related questions you might explore:

1. How is the margin of error calculated for this proportion?
2. What would be the confidence interval for the population proportion with 95% confidence?
3. How would increasing the sample size affect the confidence interval?
4. Can the normal approximation be used for any sample size, or are there restrictions?
5. How does the variability of the proportion affect the required sample size for future surveys?

**Tip:** The larger the sample size, the narrower the confidence interval, providing a more precise estimate of the population proportion.

A survey of 2302 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 406 have donated blood in the past two years. Complete parts​ (a) through​ (c) below.
Part 1​ (a) Obtain a point estimate for the population proportion of adults in the country aged 18 and older who have donated blood in the past two years.
Part 2​ (b) Verify that the requirements for constructing a confidence interval about p are satisfied.

This problem involves finding the point estimate for the population proportion of adults who have donated blood in the past two years based on a survey of 2302 adults. It also verifies if the conditions for constructing a confidence interval are met. Key concepts include population proportion and confidence intervals.

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