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Calculating and understanding the sample proportion is the foundation of creating a confidence interval for a population proportion. When we're given data, like the fact that 0.2% of 2673 individuals have both risk factors for AIDS, we use it to find the sample proportion, represented as \( \hat{p} \). This is found by dividing the percentage by 100, so 0.2% becomes 0.002. Think of this as the fraction of the sample that exhibits the characteristic we're interested in. Remember:

• The sample proportion \( \hat{p} \) tells us about the sample, not the entire population.
• A higher sample size often gives a more reliable estimate of the true population proportion.

Getting the sample proportion right is essential because it is the basis upon which further calculations are built.

The population proportion, denoted by \( p \), represents the true proportion of the entire group we're interested in. In this example, it would be the actual percentage of all adult heterosexuals who have both risk factors for AIDS. In practice, we rarely know \( p \) precisely. Instead, we estimate it using our sample proportion \( \hat{p} \). It’s like trying to guess the taste of an entire dish by sampling just a spoonful. The key here is the concept of estimation – we use sample data to make educated guesses about the larger population. This is crucial for situations where directly gathering data from the whole population is impractical.

Normal approximation plays a vital role in finding confidence intervals. This involves assuming that the distribution of the sample proportion follows a normal distribution. This assumption allows us to use statistical techniques to calculate confidence intervals, which estimate where the true population proportion might lie. For a normal approximation to hold, our sample size should be large enough, and it should meet specific conditions (you’ll see how these are checked in the Sample Size Condition section). Put simply, the concept of normal approximation simplifies calculations by allowing us to use the properties of the normal distribution. This makes it much easier to make accurate estimations about the population.

Ensuring that your sample size is adequate is critical for making valid statistical inferences. The sample size condition demands that both \( n\hat{p} \geq 5 \) and \( n(1-\hat{p}) \geq 5 \). These conditions ensure that the sample proportion approximately follows a normal distribution. In our example:

• \( n\hat{p} = 5 \) confirms that the number of individuals with the characteristic is sufficient.
• \( n(1-\hat{p}) = 2668 \) confirms that the number of individuals without the characteristic is also sufficient.

Meeting these conditions gives us confidence that we can reasonably approximate the population's characteristics based on our sample. It is a way to check that our sample is robust enough to provide reliable insights into the population, making the analysis outcomes meaningful and trustworthy.