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The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It's measured in terms of standard deviations from the mean. If a Z-score is 0, it means the value is exactly average; if it's positive, the value is above average; and if it's negative, the value is below average.

To compute a Z-score, use the following formula:
\[\begin{equation} Z = \frac{X - \bar{\text{X}}}{\text{\text{σ}}} \end{equation}\]
Here,

• X is the value you're evaluating,
• \bar{\text{X}} is the mean of all values, and
• σ is the standard deviation.
  

For instance, in this exercise, if X = 65, the mean (µ) = 50, and the standard deviation (σ) = 7, the Z-score calculates to approximately 2.14. This means the value 65 is 2.14 standard deviations above the mean.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wide range.

To compute standard deviation, use the formula:
\[\begin{equation} \text{σ} = \frac{1}{N} \times \text{Σ} \times \text{( X - \bar{\text{X}} )}^{2}\end{equation}\]

• N is the number of values,
• X represents the individual values,
• X-bar is the mean of the values.
• Σ is the summation symbol, meaning to sum up the squared differences.
  

For example, in this scenario, where the standard deviation σ = 7, it tells us how much the values in the dataset deviate, on average, from the mean value of 50.

Cumulative probability is the probability that a random variable is less than or equal to a certain value. In the context of the standard normal distribution, it's the area under the curve to the left of a given Z-score.

To find this in a Z-table, locate the Z-score row and column matching your value. The intersection of these row and column gives you the cumulative probability.

For example, with a Z-score of 2.14, the cumulative probability P(Z < 2.14) = 0.9838. This means there is a 98.38% chance a value chosen randomly from the distribution will be less than 2.14 standard deviations above the mean. To find P(Z > 2.14), we subtract this value from 1 as illustrated in the exercise:
\[\begin{equation}P(Z > 2.14) = 1 - P(Z < 2.14) = 1 - 0.9838 = 0.0162\end{equation}\] This results in a 1.62% chance of a value being higher than 2.14 standard deviations above the mean.

The normal curve, or the Gaussian distribution, is a symmetric, bell-shaped curve that shows the distribution of a dataset. The highest point on the curve is at the mean, and the curve tails off symmetrically on both sides.

Key Characteristics:

• Symmetry: The curve is symmetric around the mean.
• Mean, Median, Mode: In a perfectly normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.
• 68-95-99.7 Rule: About 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
  

When we draw the normal curve for the exercise, we mark 50 at the center (mean), then shade the area to the right of 65. This shaded region represents P(X > 65), showing the probability that a value will fall to the right of the Z-score of 2.14.