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The population proportion, denoted as \( p \), is a fundamental concept in statistics. It represents the true proportion of individuals in the entire population that possess a specific characteristic. In our context, we want to estimate the proportion of adults in the entire population who have both received a blood transfusion and have a high-risk sexual partner.
Understanding the population proportion is essential because we often only have access to a sample, not the entire population. As such, we aim to estimate \( p \) using data available from the sample. It serves as our best educated guess for what the actual proportion might be in the full population.
Given the exercise, since the entire population cannot feasibly be surveyed, a sample analysis provides a practical solution for estimation.

The sample proportion, denoted as \( \hat{p} \), is our calculated proportion based on the sample data. It provides a direct estimate of the population proportion.
In our case, we determined \( \hat{p} \) by considering a sample of 2673 adult heterosexuals. Out of these, around 0.2% had both characteristics of interest. By calculating \( \hat{p} = \frac{0.2\%}{100} \times 2673 \), we simplified it to the number of individuals, yielding \( \hat{p} = \frac{5}{2673} \approx 0.00187 \).
This sampling gives us a snapshot of the larger population. However, since it is based on a sample, there will always be some degree of uncertainty involved, which is why confidence intervals are necessary.

To calculate a confidence interval for a population proportion accurately, certain statistical conditions must be met. These conditions ensure that our estimate is reliable and meaningful. Specifically, the focus is on the sample size and the number of successes in the sample.
Two key conditions must be satisfied:

• The product of the sample size and the sample proportion (\( n\hat{p} \)) must be at least 5.
• The product of the sample size and the complement of the sample proportion (\( n(1-\hat{p}) \)) must also be at least 5.

In our exercise, we found:

• \( n\hat{p} = 2673 \times 0.002 = 5.346 \),
• \( n(1-\hat{p}) = 2673 \times 0.99813 = 2667.65 \).

Both values are above or equal to 5, meaning these conditions are met. This verification is crucial for ensuring the appropriateness of proceeding with constructing a confidence interval.

Sample size is a critical component in inferential statistics and significantly impacts the precision of the confidence interval. When estimating a population proportion, the sample size (denoted as \( n \)) influences the variability of the sample proportion. A larger sample size typically results in a more accurate estimate of the population proportion.
In the given exercise, our sample size is 2673 which is relatively large. This large sample size helps to decrease the margin of error in our estimation and increases the likelihood that our sample proportion is representative of the population.
It's important to note that while a large sample size is beneficial, it must also be manageable and practical for the situation. However, having a sample size that is too small might lead to unreliable estimates. Hence, ensuring an appropriately sized sample is essential for meaningful statistical analysis.