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Understanding the Z-score is crucial when dealing with normally distributed data, like the wages in the given exercise.
A Z-score, or standard score, indicates how many standard deviations an element is from the mean. In this case, it helps determine how an employee's wage compares to the average wage of the corporation.

For any given value, the Z-score is calculated using the formula: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value (wage), \( \mu \) is the mean average wage, and \( \sigma \) is the standard deviation of the wages. If a Z-score is 0, it signifies that the wage is exactly the average. A positive Z-score indicates a wage above the average, while a negative score points to a wage below the average.

To improve understanding, let's look at how this applies to the exercise:

• For wages of \(11.00 and \)14.00, you would subtract the mean (\(14.50) from each and then divide by the standard deviation (\)1.50).
• For a wage of $16.00 to answer part (b), you would similarly calculate its Z-score to understand how it stands in relation to the average.

While Simpson's Rule was mentioned in the exercise, it is actually not used for the standard normal distribution case presented. However, Simpson's Rule is an important numerical method for estimating the definite integral of a function.

\textbf{What is Simpson's Rule?}
Simpson's Rule is used to estimate the area under a curve, which is useful for approximating definite integrals. It employs parabolas to approximate each segment of the curve between specified intervals. The formula for Simpson's Rule is: \( I \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)] \), where \( I \) is the integral approximation, \( \Delta x \) is the width of each interval, and \( f(x_i) \) represents the function value at the ith interval.

To connect this to real-world applications, imagine using Simpson's Rule to determine the flow rate of a river or the amount of fuel a rocket needs for a space mission—both situations where the function's exact integral is complex or unknown.

The standard normal distribution is a critical concept in statistics, providing a basis for comparison for normally distributed variables. It is a special case of the normal distribution with a mean \( \mu = 0 \) and a standard deviation \( \sigma = 1 \). In a standard normal distribution, the Z-scores directly give us the position in relation to the mean.

In the context of the exercise, once you calculate the Z-scores for the given wages, you can refer to the standard normal distribution to find out what percentage of employees have wages within those Z-score ranges.

Here’s how it works:

• The total area under the standard normal distribution curve equals 1, or 100% when considering probabilities.
• The area to the left of a Z-score gives the probability that a value is less than or equal to that Z-score.
• For wages, this translates to the chance of an employee earning up to a certain amount.

Using standard normal distribution tables, or specific functions in statistical software, allows users to determine these probabilities accurately, as was done in step 2 and step 3 for parts (a) and (b) of the problem respectively.