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To understand a sample proportion, let's first break down the survey. We start by looking at the responses to see how many managers said 'yes,' meaning they usually take work home. Add up all the 'Yes' answers from the given responses to find the total number. In this example, there are 8 managers who answered 'Yes'. This number is then divided by the total number of managers surveyed, which is 20 in this exercise. This gives us the sample proportion, represented by the symbol \( p \). So, here, \( p = \frac{8}{20} = 0.4 \), or 40%. This means that in the sample, 40% of the managers usually take work home.
Understanding the sample proportion is crucial because it helps estimate the proportion of a larger population, based on a small sample. In statistics, the sample proportion is often used to make predictions about an entire population's behavior or characteristics.

The standard error is a measure that tells us how much the sample proportion might vary from the actual population proportion. It gives an idea of the uncertainty or the potential error in our estimate.
To calculate the standard error for a proportion, use the formula: \( SE = \sqrt{\frac{p(1 - p)}{n}} \). This formula requires the sample proportion \( p \) and the total number of responses \( n \).
In our example, with \( p = 0.4 \) and \( n = 20 \), the calculation would be:

• \( SE = \sqrt{\frac{0.4 \times (1 - 0.4)}{20}} \)

This standard error helps us understand the variability of the sample, offering insight into how well the sample proportion represents the larger population. The smaller the standard error, the closer the sample proportion is likely to be to the population proportion.

The Z-table critical value is a number from the Z-table that helps set the bounds for a confidence interval. It represents the number of standard deviations a data point is from the mean in a normal distribution. Choosing the right Z-value depends on the level of confidence we want to have in our interval estimate.
For a 99% confidence interval, as noted in the exercise, the critical value is 2.58. This high Z-value reflects the high level of certainty we want when estimating the population proportion.
To use the critical value, multiply it by the standard error, then add and subtract the result from the sample proportion \( p \) to find the confidence interval. This gives you the range within which the true population proportion is likely to fall with 99% confidence. The calculation looks like this:

• Confidence interval = \( p \pm (Z \times SE) \)

The larger the Z-value, the wider the confidence interval, reflecting increased certainty regarding the estimate, even as the specific population numbers remain uncertain.