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The normal curve, also known as the Gaussian curve, is a common way of representing a normal distribution. This curve is symmetric about the mean, which is located at the center of the distribution. It has a bell shape and its highest point occurs at the mean. The spread of the curve is determined by the standard deviation.

• The mean marks the central point.
• The standard deviation measures how spread out the values are.

In the given exercise, the normal curve is centered at 50 with a standard deviation of 7. When you draw this curve, you will see the bell shape centered at 50, and your task is to find the probability that a value falls to the left of 45.

The Z-score, or standard score, describes the position of a particular value in terms of standard deviations from the mean. It allows us to prescribe how unusual or typical a certain value is within a normal distribution.
The Z-score is calculated using the formula: \ \(Z = \frac{X - \mu}{\sigma}\).
Here's what each part means:

• \mu is the mean of the dataset.
• \sigma is the standard deviation.
• X is the value you are converting to a Z-score.

In the problem, we computed the Z-score for X = 45 which corresponds to \ \( Z \approx -0.71 \). This tells us that 45 is about 0.71 standard deviations below the mean.

A standard normal variable (Z) is a normalized version of a random variable, adjusted so that it has a mean of 0 and a standard deviation of 1. This standardization allows for easy comparison across different datasets.
The process of standardizing converts any normal distribution to a standard normal distribution using the calculated Z-score.

• A Z-score of 0 corresponds to the mean.
• Positive Z-scores are above the mean.
• Negative Z-scores are below the mean.

With a Z-score of \ \( Z \approx -0.71 \), we've determined where our value (45) falls relative to the mean (50) of our original distribution. The next step is to use the Z-score to find the desired probability.

Once you have the Z-score, the next step is to calculate the probability. This is done using a Z-score table or an online calculator.
These tools provide the area under the standard normal curve to the left of the given Z-score. For our Z-score of \ \( Z \approx -0.71 \), this area, or probability, is approximately 0.2389. This means that there is a 23.89% chance that a value will fall to the left of 45 in our original distribution.

By sketching the normal curve, labeling the mean, and shading the appropriate area, you can visually represent the solution. This helps in understanding how the components of the normal distribution and Z-scores work together to calculate probabilities.