Part 1: Obtain a Point Estimate for the Population Proportion

The point estimate for the population proportion, $\hat{p}$, is given by:
$\hat{p} = \frac{\text{Number of successes (donated blood)}}{\text{Total number of adults surveyed}}$

Substitute the values:
$\hat{p} = \frac{412412}{22832283} \approx 0.018$

Answer for Part (a):
$\hat{p} = 0.018$ (rounded to three decimal places)

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Part 2: Verify Requirements for Constructing Confidence Interval

The requirements for constructing a confidence interval about $p$ are:

1. The sample can be reasonably assumed to be a simple random sample.
2. The value of $n \hat{p}(1 - \hat{p})$ must be greater than or equal to 10.
3. The population size must be at least 20 times the sample size (equivalently, the sample size must be less than or equal to 5% of the population).

Step 1: Simple Random Sample

The problem states this is a survey conducted by a reputable polling organization. Hence, it is reasonable to assume the sample is a simple random sample.

Step 2: Check $n \hat{p}(1 - \hat{p})$

$n \hat{p}(1 - \hat{p}) = 22832283 \times 0.018 \times (1 - 0.018)$ $n \hat{p}(1 - \hat{p}) = 22832283 \times 0.018 \times 0.982 \approx 402538$ Since $402538 \geq 10$, this condition is satisfied.

Step 3: Population Size Relative to Sample Size

Assume the population size of adults in the country is significantly larger than the sample size $n = 22832283$. Thus, the sample size is less than 5% of the population, satisfying this condition.

Answer for Part (b):

• The sample can be assumed to be a simple random sample.
• The value of $n \hat{p}(1 - \hat{p}) = 402538$, which is greater than or equal to 10.
• The sample size is less than or equal to 5% of the population size.

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Part 3: Construct and Interpret a 90% Confidence Interval

The formula for a confidence interval for a population proportion is:
$\hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$
For a 90% confidence level, the critical value $z = 1.645$.

Step 1: Calculate the Margin of Error (ME)

$\text{ME} = z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ $\text{ME} = 1.645 \sqrt{\frac{0.018 \times (1 - 0.018)}{22832283}}$ $\text{ME} = 1.645 \sqrt{\frac{0.018 \times 0.982}{22832283}} = 1.645 \sqrt{\frac{0.017676}{22832283}}$ $\text{ME} = 1.645 \times \sqrt{7.741 \times 10^{-10}} \approx 1.645 \times 0.000879 \approx 0.001445$

Step 2: Compute Confidence Interval

$\text{Lower bound} = \hat{p} - \text{ME} = 0.018 - 0.001445 \approx 0.0166$ $\text{Upper bound} = \hat{p} + \text{ME} = 0.018 + 0.001445 \approx 0.0194$

Answer for Part (c):

• We are 90% confident the proportion of adults aged 18 and older in the country who have donated blood in the past two years is between 0.017 and 0.019.

Correct Choice:
A. We are 90% confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between 0.017 and 0.019.

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Follow-up Questions:

1. What does the margin of error indicate about the precision of the estimate?
2. How would the confidence interval change with a higher confidence level, like 95%?
3. Why is it important to verify the conditions before constructing a confidence interval?
4. How can sampling bias affect the reliability of this confidence interval?
5. How would a larger sample size influence the margin of error?

Tip: Confidence intervals give a range of plausible values for the true population proportion but do not guarantee the exact proportion lies within the interval.