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The sample proportion is a key component when calculating confidence intervals in statistics. It represents the fraction of the surveyed subject that exhibits the characteristic of interest. In the current exercise, the sample proportion tells us what part of the homes surveyed met the minimal earthquake preparedness guidelines. To find it, you divide the number of homes that met the criteria by the total number of homes surveyed.For example:

• Total homes surveyed: 511
• Homes meeting guidelines: 173

By dividing 173 by 511, we get the sample proportion, often denoted as \( \hat{p} \). So in this exercise, \( \hat{p} = \frac{173}{511} \approx 0.338 \). This proportion helps us understand how widespread the preparedness is among the surveyed homes. The sample proportion is always a number between 0 and 1, where a higher number indicates a larger percentage of homes meeting the requirements.

The standard error plays a crucial role in estimating how much uncertainty surrounds your sample proportion. It measures how much the sample proportion would vary if you were to take multiple samples from the same population, and it's essential for constructing a confidence interval.To calculate the standard error (SE) of a sample proportion, you use the formula:\[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]where \( \hat{p} \) is the sample proportion, and \( n \) is the sample size.In our exercise:

• \( \hat{p} = 0.338 \)
• \( n = 511 \)

Plugging these values into the formula:\[SE = \sqrt{\frac{0.338 \times 0.662}{511}} \approx 0.0215\]The standard error gives us an idea of the variability and helps us gauge how reliable our sample proportion is for estimating the true population proportion.

The margin of error enables us to understand the potential range of error in our sample estimate. It is calculated by multiplying the standard error by the Z-score that corresponds to your desired confidence level. This gives us a range above and below the sample proportion, which we use to construct the confidence interval.To determine the margin of error (ME):

• Use the formula: \[ME = Z \times SE\]
• For a 90% confidence interval, the Z-score is approximately 1.645.
• With a standard error of approximately 0.0215, the calculation becomes:\[ME = 1.645 \times 0.0215 \approx 0.0354\]

This margin of error tells us that the sample proportion could reasonably vary by about \( \pm 0.0354 \) due to sampling variability. It's a crucial step in forming the boundaries of the confidence interval and gives us a clearer picture of the uncertainty involved.

The Z-score is a statistical measure that tells us how many standard deviations an element is from the mean. In terms of confidence intervals, it is used to determine the critical value for a specific confidence level. It acts as a multiplier for the standard error when calculating the margin of error.For a 90% confidence level:

• The Z-score is 1.645.
• This value comes from the standard normal distribution table.

The Z-score factors into the formula for the margin of error: \[ME = Z \times SE\]. This helps in adjusting the width of the confidence interval.Using a Z-score of 1.645, the confidence interval becomes a more reliable range for estimating the true population proportion. Understanding the Z-score is vital for interpreting the confidence interval correctly and grasping the confidence we have in our estimate.