Short Answer 
The problem involves a sample size of 500 with a population proportion of 0.47. After calculating the mean and standard deviation, the probability that the sample proportion exceeds 0.50 is found to be approximately 0.089264 using the z-score method.
Step 1: Understand the Given Data
In the problem, we are provided with several important details necessary for calculating the desired probability. Specifically:
- Sample Size (n): 500
- Population Proportion (p): 0.47
These values will be used to calculate the mean and standard deviation of the sampling distribution to help find the probability.
Step 2: Calculate the Mean and Standard Deviation
The next step involves calculating the mean and the standard deviation of the sampling distribution using the formulas:
- Mean ((mu_x)): (mu_x = p = 0.47)
- Standard Deviation ((sigma)): (sigma = sqrt{frac{p(1-p)}{n}} = sqrt{frac{0.47(1-0.47)}{500}} = 0.0223)
These calculations are vital as they will be used in determining the probability that the sample proportion exceeds 0.50.
Step 3: Find the Required Probability
We can now compute the probability that the sample proportion (P(X > 0.50)) using the z-score formula:
- Z-Score Calculation: (P(X > 0.50) = Pleft(frac{X – mu}{sigma} > frac{0.50 – 0.47}{0.0223}right) = P(Z > 1.3453))
- Z-Table Value: The area under the normal curve to the left of (Z = 1.3453) gives us (P(Z > 1.3453) = 0.089264)
Thus, the final result for the probability that more than 50% of the sample supports the measure is (P(X > 0.50) = 0.089264).