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In statistics, a **proportion** refers to a part or fraction of a whole. It is often expressed as a percentage or a decimal. When calculating the proportion in a given sample, you divide the count of the specific interest by the total number in the sample.
For instance, in the ice-skating class example, the proportion of girls is calculated as the number of girls divided by the total number of students in the class.
Steps to find the proportion:

• Identify the count of the subgroup of interest (e.g., girls in the class — 64).
• Determine the total sample size (e.g., total children in the class — 80).
• Use the formula: \( p^{\prime} = \frac{\text{subgroup count}}{\text{total count}} \).

For the given class, \( p^{\prime} = \frac{64}{80} = 0.8 \), meaning 80% of the class are girls.

The **standard error** (SE) measures how much the sample proportion is expected to vary from the true population proportion. It reflects the precision of the sample proportion as an estimate of the population proportion.
For a proportion, the standard error is calculated with the formula:

• \( SE = \sqrt{\frac{p^{\prime}(1-p^{\prime})}{n}} \)

Where \(p^{\prime}\) is the sample proportion and \(n\) is the sample size.
In our ice-skating class example, \( p^{\prime} = 0.8 \) and \( n = 80 \), so:

• \( SE = \sqrt{\frac{0.8(0.2)}{80}} = \sqrt{\frac{0.16}{80}} \approx 0.0447 \)

A smaller standard error indicates a more precise estimate.

A **Z-score** tells us how many standard deviations an element is from the mean in a normal distribution. It is crucial in determining confidence intervals. For instance, it helps in adjusting the width of the confidence interval proportionate to the desired confidence level.
To find an appropriate Z-score for a given confidence level, refer to a Z-table or use a statistical calculator.
For a 98% confidence interval, the corresponding Z-score is approximately 2.33. This means we expect the true proportion to lie within 2.33 standard deviations from the sample proportion and the population mean. A higher Z-score reflects a higher confidence level, thus increasing the interval width.

The **error bound** is the additional amount we add and subtract from the sample proportion to create a confidence interval, expressing the uncertainty or margin of error in our estimate. It depends significantly on the Z-score and standard error.
Calculate the error bound margin using:

• \( EBM = Z \cdot SE \)

In our example, applying a Z-score of 2.33 for 98% confidence level and an SE of approximately 0.0447:

• \( EBM = 2.33 \times 0.0447 \approx 0.1042 \)

By increasing the confidence level from a lower percentage to 98%, the Z-score, and consequently the error bound, also increase. This suggests the confidence interval becomes wider, offering a higher chance of containing the true population parameter.