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In the context of survey statistics, the sample proportion is a measure used to estimate the proportion of a certain characteristic within a population. To calculate the sample proportion, you would divide the number of individuals exhibiting the characteristic of interest by the total number of individuals surveyed. For example, in our exercise, a total of 300 employed individuals were surveyed, and 35 of them reported working from home at least once per week. Therefore, the sample proportion, denoted as \( \hat{p} \), can be computed as follows: \ \[ \hat{p} = \frac{35}{300} = 0.1167 \ \]. This means that approximately 11.67% of the sampled employees work from home at least once per week. Calculating the sample proportion is the first essential step in constructing a confidence interval.

The standard error of a sample proportion measures the variability or precision of the sample proportion as an estimate of the population proportion. It tells us how much the sample proportion might differ from the true population proportion. The smaller the standard error, the more precise our sample proportion estimate is.
To calculate the standard error, we use the formula: \[ \text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \ \].
Here, \ \( \hat{p} \) is our sample proportion (0.1167) and \ \( \text{n} \) is the sample size (300). Plugging in these values, we get: \[ \text{SE} = \sqrt{\frac{0.1167 \(1 - 0.1167 \)}{300}} \ \approx 0.0189 \].
This result indicates that the standard error for our sample proportion is 0.0189, and it will be used to calculate the margin of error next.

The margin of error helps to determine the range within which the true population proportion is likely to fall. It provides us with a measure of uncertainty around our sample proportion estimate. To calculate the margin of error, we use the formula: \ \[ \text{ME} = Z \times \ \text{SE} \ \].
Here, \ \( Z \) is the Z-value corresponding to our desired confidence level, and \ \( \text{SE} \) is the standard error calculated in the previous step. For a 99% confidence interval, the Z-value is approximately 2.576.
Plugging in the values, we get: \[ \text{ME} = 2.576 \times 0.0189 \]
\[ \text{ME} \ \approx 0.0487 \]
This margin of error means that there is a predicted range of error of about 4.87% in our sample proportion estimate.

The Z-value, also known as the Z-score, is a critical value derived from the standard normal distribution. It represents the number of standard deviations a data point is from the mean. In the context of confidence intervals, the Z-value corresponds to the desired level of confidence that the true population proportion lies within the calculated interval.
For a 99% confidence interval, the Z-value is approximately 2.576. This value can be found in Z-tables, which provide the critical values for different confidence levels. Here's why it is important:

• The Z-value ensures that our interval has the correct level of confidence.
• It reflects how rare or common the sample outcome is in the context of the normal distribution.
It is crucial to use the Z-value accurately so that we reliably capture the desired confidence level in our interval.
Using this Z-value along with the standard error, we can determine the margin of error, which then allows us to construct our confidence interval.