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The z-score is a measure that tells you how many standard deviations a data point is from the mean. In the context of the standard normal distribution, the mean is always 0, and the standard deviation is 1.
This means that a z-score tells you where a particular value lies on a standard normal distribution curve. For example, if you have a z-score of 2, the data point is 2 standard deviations above the mean.Calculating a z-score can be done with the formula:

• \( z = \frac{X - \mu}{\sigma} \)

Here, \( X \) is your data point, \( \mu \) is the mean of the dataset, and \( \sigma \) is the standard deviation. In the exercise case, you're already given a value of \( z = 3.20 \) which sits on the curve according to the standard normal distribution.Z-scores are crucial when you're trying to understand where an observation sits relative to the entire dataset, especially in relation to probabilities. They allow you to transform normal data to a standard scale.

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain value. In the context of z-scores, it's the probability of a z-score being less than or equal to a given value.
This value can be obtained from a z-table, which contains cumulative probabilities associated with z-scores. For example, to find the probability that a z-score is less than or equal to 3.20, you would look up 3.20 in a z-table. The table tells you the area under the standard normal curve to the left of that z-score.
In our given scenario, this probability is approximately 0.9993. Understanding cumulative probability helps you determine the likelihood of a random variable falling within any particular region of the distribution, which is vital for making statistical inferences.

• Helps assess probabilities over specific intervals.
• Useful in hypothesis testing and confidence interval calculations.

The standard normal curve is a graphical representation of the standard normal distribution, which has a mean of 0 and a standard deviation of 1. When data is graphed, it forms the familiar bell-shaped curve.The curve itself is symmetric about the mean. This symmetry means that around 68% of data falls within 1 standard deviation of the mean (between z-scores of -1 and 1), about 95% within 2 standard deviations, and around 99.7% within 3 standard deviations. This is known as the empirical rule.The standard normal curve is a staple in statistics because it provides a baseline for comparing different sets of data. By converting data into z-scores, you can determine how likely or unlikely certain outcomes are, based on their position on the curve.In this exercise, when you calculate \( P(z \leq 3.20) \), you are essentially shading the area under the standard normal curve to the left of the 3.20 z-score. This shaded region visually represents the probability that a z-score will be less than or equal to 3.20, which is the area of 0.9993 found in the exercise.