91Ó°ÊÓ

When we want to estimate a proportion, like how many people have blood type B, it's crucial to calculate the right sample size. The sample size affects how confident we can be about our estimate.
In the absence of a preliminary estimate of the population proportion \( p \), a common approach is to use the maximum variability assumption. This assumes \( p = 0.5 \) because 0.5 reflects the greatest uncertainty (and variability) in a binary outcome. This way, the estimated sample size is less prone to underestimation.
For our problem, the calculation formula is:

• \[ n = \left(\frac{Z^2 \cdot 0.25}{E^2}\right) \]

• \( n \) is the sample size we want to find.
• \( Z \) is the Z-score linked to the confidence level (1.44 for 85%).
• \( E \) is the margin of error, being 0.05 in our example.

This formula considers the maximum potential error (0.25 derived from \( 0.5 \) times \( 0.5 \)) for scenarios without prior proportion estimates.

Proportion estimation is a statistical technique used to infer information about a population from a sample. Imagine we want to estimate how many people in a city have a certain blood type. We can't test everyone, so we use a representative sample.
In our exercise, when you have a preliminary estimate, such as finding 8 blood type B individuals out of 90, it helps refine further calculations. To make use of a proportion estimate, use the formula:

• \[ \hat{p} = \frac{8}{90} = 0.0889 \]

This \( \hat{p} \) serves as a starting point to narrow down variability, which impacts the required sample size. When you base calculations on this preliminary estimate,

• the new formula becomes: \[ n = \left(\frac{Z^2 \cdot \hat{p} \cdot (1 - \hat{p})}{E^2}\right) \]

By substituting our \( \hat{p} \) value, the sample size needed significantly reduces because it reflects the population more accurately.

The Z-score is essential in estimating how confident we can be about our results. It indicates how many standard deviations an element is from the mean. In the realm of confidence intervals, a higher Z-score means more confidence.
For example, an 85% confidence level implies that we're 85% sure our estimate falls within a specified range. To determine the right Z-score for this level, it's essential to find the values corresponding to the tails of the standard normal distribution. For an 85% confidence level:

• The central part holds 85%, leaving 7.5% in each tail.
• Using a Z-table, a 7.5% tail equates to a Z-score of approximately 1.44.

Thus, for our calculations, \( Z = 1.44 \), helping us ensure our results are within bounds of reliability. Understanding how the Z-score supports confidence is important for correctly interpreting the level of certainty in statistical outcomes.