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When determining the confidence interval for a population proportion, the first step is identifying the sample proportion. This sample proportion, denoted as \( \hat{p} \), is a measure of what proportion of our sample possesses the characteristic of interest. In the exercise, the characteristic is owning at least one firearm. To calculate \( \hat{p} \), we use the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successful outcomes (households with a firearm) and \( n \) is the total number of observations (total households surveyed).
Applying this formula in our example:
- \( x = 133 \) (households owning firearms)
- \( n = 539 \) (total households)
Therefore, \( \hat{p} = \frac{133}{539} \approx 0.2467 \).
This means approximately 24.67% of the sampled households own at least one firearm, and this proportion will serve as the basis for further calculations.

The Z-score plays a crucial role in confidence interval calculations. It helps to determine how far and in what direction the sample proportion \( \hat{p} \) should be adjusted to estimate the population proportion with a specified confidence level. The Z-score is a statistical measure that represents the number of standard deviations a data point is from the mean.
For a 95% confidence level in a one-tailed context (as we are only interested in the lower bound), we use a Z-score of 1.645. This implies that on a standard normal distribution, 95% of the data lies below this score.

• The value of 1.645 comes from standard Z-score tables, which provide the probability of data points falling within a certain range.
• This specific Z-score is essential in calculating how much we adjust the sample proportion to find the lower bound for our confidence interval.

With the Z-score in hand, we move forward to calculate the standard error.

Standard error (SE) measures the variability or dispersion of the sample proportion. It essentially tells us how much the sample proportion \( \hat{p} \) would vary if we took multiple samples from the population. The formula for SE when dealing with proportions is: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]Substituting the values from our exercise gives us:

• \( \hat{p} = 0.2467 \)
• \( n = 539 \)

So our SE calculation becomes:\[ SE = \sqrt{\frac{0.2467 \times (1 - 0.2467)}{539}} \approx 0.0194 \]A smaller SE implies that our sample proportion is a more accurate estimate of the population proportion. The SE is used in the next step to calculate the confidence bound.

The lower confidence bound gives us a range within which we can be confident that the actual population proportion will fall, considering only the lower limit. It is calculated using the sample proportion, the Z-score, and the standard error. For the lower bound, the formula is:\[ \text{Lower Bound} = \hat{p} - Z \times SE \]Using the values obtained earlier in the exercise, we have:

• \( \hat{p} = 0.2467 \)
• \( Z = 1.645 \)
• \( SE = 0.0194 \)

Therefore, the calculation becomes:\[ 0.2467 - 1.645 \times 0.0194 \approx 0.2149 \]This result means that we can be 95% confident that the true proportion of households owning at least one firearm in the city is at least 21.49%. This lower bound helps in understanding the minimum scope of firearm ownership, providing a basis for further analysis or action.