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Z-scores are a way to describe the position of a value within a normal distribution. They tell you how many standard deviations away from the mean a particular value lies. We use the formula \[Z = \frac{(X - \mu)}{\sigma}\] where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
In the given example, the mean is 50 and the standard deviation is 7. To find the z-score for \(X = 55\):
\[Z_1 = \frac{(55 - 50)}{7} \approx 0.714\] This means that 55 is approximately 0.714 standard deviations above the mean.
Similarly, for \(X = 70\):
\[Z_2 = \frac{(70 - 50)}{7} \approx 2.857\] This indicates that 70 is around 2.857 standard deviations above the mean.

A standard normal distribution table helps us find the probability that a standard normal random variable is less than or equal to a z-score. The table provides the area to the left of the z-score under the standard normal curve.
In the example, after finding the z-scores, we look up \(Z_1 = 0.714\) and \(Z_2 = 2.857\) in the table.
For \(Z_1 = 0.714\), we find:
\[P(Z \leq 0.714) \approx 0.7625\]
This means there is a 76.25% probability that a value is less than or equal to 0.714 standard deviations above the mean.
For \(Z_2 = 2.857\), the table shows:
\[P(Z \leq 2.857) \approx 0.9978\]
This indicates a 99.78% probability that a value is less than or equal to 2.857 standard deviations above the mean.

Visualizing the problem with a normal curve is very important. It helps you understand the concept of probabilities in a distribution.
Once you have your z-scores, you can shade the area under the curve between them to represent the probability.
In the exercise, we are interested in \(P(55 \leq X \leq 70)\). We've already found the corresponding z-scores \(Z_1 = 0.714\) and \(Z_2 = 2.857\).
To find this probability, subtract the smaller area from the larger area:
\[P(55 \leq X \leq 70) = 0.9978 - 0.7625 \approx 0.2353\]
This is the shaded area under the curve between these z-scores.
Drawing the normal curve with the shaded area highlights the probability and makes it easier to understand.