Short Answer 
The normal distribution is used to analyze voter support for a ballot measure, with a sample of 500 voters showing a proportion of 0.47. The Z-score for a target proportion of 0.50 is calculated to be 1.34, corresponding to a p-value of 0.9099, indicating a 9.01% chance that more than half the sampled voters support the measure.
Step 1: Understand the Basics of the Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by mean (μ) and standard deviation (σ). In this context, we are examining voter support for a ballot measure in Region A. Using the Central Limit Theorem, we can analyze sample proportions, as these will tend to follow a normal distribution as the sample size increases. For our sample of 500 voters, we have a proportion (p) of 0.47.
Step 2: Calculate the Z-Score
The Z-score is a statistical measurement that describes a value’s relationship to the mean of a group of values. It is calculated with the formula Z = (X – μ) / σ. Here, we want to find the Z-score for when X (the proportion of voters supporting the measure) is 0.50. First, we compute the standard deviation (σ) using the formula σ = ‚à ö(n * p * (1 – p)), which gives σ = 0.0223 for our parameters. Plugging these numbers into the Z-score formula, we find Z = (0.50 – 0.47) / 0.0223 = 1.34.
Step 3: Find the Probability and Interpret the Results
Once we have the Z-score, we consult the Z-score table to find the corresponding p-value, which indicates the probability of observing a value less than or equal to X. For Z = 1.34, the p-value is 0.9099, which translates to approximately 90.99% of voters supporting 0.50 or less. To find the probability of support being greater than 0.50, we calculate 1 – p-value, yielding a final probability of 0.0901, or 9.01%. This indicates the chance that more than half of the sampled voters support the measure.