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In statistics, the confidence level is a key concept that tells us how certain we can be about the accuracy of our sample results. It is often expressed as a percentage, such as 95% or 99%. The confidence level indicates the probability that the confidence interval contains the true population parameter. For example, a 95% confidence level means we are 95% certain that our sample proportion is within the margin of error of the true population proportion.
The confidence level is used to determine the z-score in margin of error calculations. Z-scores are a number from the standard normal distribution. At a 95% confidence level, the z-score is approximately 1.96.
The higher the confidence level, the more certain we can be about our results, but this also usually results in a larger margin of error. Conversely, a lower confidence level results in a smaller margin of error but less certainty about our results.

The sample proportion is a crucial concept in statistics, representing the fraction of the sample that presents a particular characteristic. For example, in the problem, 39% (or 0.39) of the people polled said they have guns. This 39% is the sample proportion. It helps us estimate the proportion of the population that shares the same characteristic but within a margin of error.
The sample proportion is used in various calculations, including determining the margin of error and establishing confidence intervals. In our case, it's part of the formula to calculate the margin of error and to solve for the number of people surveyed (n). The formula is:
\( E = z \times \frac{\root \frac{\text{p} \times (1 - \text{p})}{n}} \)
Here, E is the margin of error, z is the z-score, p is the sample proportion, and n is the number of people surveyed.

A z-score is a statistical measurement that describes a value's relationship to the mean. In the context of confidence intervals and margin of error calculations, the z-score represents how many standard deviations a point is from the mean of the distribution. The z-score is based on the desired confidence level.
For example, a 95% confidence level has a corresponding z-score of approximately 1.96, which is used in our margin of error formula. This score is derived from the standard normal distribution. The higher the z-score, the wider the confidence interval, but with a higher certainty.
In the margin of error formula:
\[ E = z \times \frac{\root \frac{\text{p} \times (1 - \text{p})}{n}} \]
The z-score helps to scale the error based on the confidence level. This is crucial for accurately calculating the margin of error and the number of people surveyed for specific confidence levels.