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When examining sample proportions, like the case with the 200 Americans and their voting opinions, the Central Limit Theorem (CLT) is pivotal. It tells us that if we take many samples from a population, the sampling distribution of the sample proportion will tend to normality as the number of samples increases, even if the original population distribution is not normal. For sizable samples (usually n > 30), the sample means are distributed nearly normally, regardless of the population's shape. This is incredibly useful because it allows us to make inferences about the population using the properties of the normal distribution, such as using Z-scores to determine probabilities.
In the example given, the sample size of 200 is well above 30, which means that, according to the CLT, the sampling distribution will closely resemble a normal distribution. This substantiates the steps taken in b) to calculate the probability using methods that are suitable for normal distributions.

Standard deviation is a measure that tells us how spread out the values in a set of data are. In the context of our problem, we are interested in the standard deviation of the sampling distribution of the sample proportion. This standard deviation is also referred to as the standard error. It provides an idea of how much we can expect a sample proportion to vary from the population proportion.
To calculate the standard deviation (or standard error) for the sample proportion, we use the formula \[\sigma = \sqrt{\frac{p(1 - p)}{n}}\]where \(p\) is the population proportion, and \(n\) is the sample size. Knowing this helps us understand the variability of the sample proportions around the population proportion of 59%. This step is crucial before proceeding to Z-score calculations for probability estimations.

A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would indicate a value that is one standard deviation from the mean. Z-scores may also be positive or negative, with the sign reflecting whether the value is above (positive) or below (negative) the mean.
In the context of our exercise, the Z-score tells us how many standard deviations the sample proportion of 55% is from the population proportion of 59%. The formula used for Z-score calculation is\[Z = \frac{X - \mu}{\sigma}\]Where \(X\) is the sample proportion, \(\mu\) is the population proportion, and \(\sigma\) is the standard deviation of the sampling distribution. Calculating the Z-score is a key step for determining the probability related to the sample proportion.

The sampling distribution is a probability distribution of a statistic obtained from a larger number of samples drawn from a specific population. In our exercise, we're dealing with the sampling distribution of the sample proportion. When we repeatedly take samples of size 200 from the population and calculate the proportion of each sample that wants to make voting easy, these proportions form a distribution themselves—the sampling distribution.
The beauty of the sampling distribution lies in its ability to give us a picture of how the sample proportions are spread out around the true population proportion. By applying the CLT, we assume that this distribution takes on a normal distribution shape, which simplifies our probability calculations greatly—leading to the last step in the problem-solving process: finding the cumulative distribution function.

Finally, let's discuss the cumulative distribution function (CDF), which is used to determine the probability that a random observation drawn from the distribution will be less than or equal to a certain value. In the normal distribution, the CDF is the area under the curve to the left of a certain Z-score. This function is crucial for our voting example.
Using the calculated Z-score, one can look up the probability that corresponds to that Z-score in the standard normal distribution (which is also the CDF of the Z-score) or use statistical software to find this probability. For the exercise, we want the probability that the sample proportion is at least 55%. Since the CDF gives us the probability of being less than a certain value, we subtract the CDF from 1 to find the percentage of times the sample proportion is greater than 55%, thus completing our probability determination.