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A z-score is a statistical measure that tells us how many standard deviations an element is from the mean of its distribution. Imagine it as a way to put a specific number within a data set into perspective with the entire set. It's commonly used in statistics to determine how normal an observation is, compared to a standard distribution like the normal distribution.

In our exercise, we used z-scores to find probabilities under a normal distribution. You calculate a z-score with the formula: \[ Z = \frac{X - \mu}{\sigma} \] where:

• \(X\) is the value in the distribution,
• \(\mu\) is the mean, and
• \(\sigma\) is the standard deviation.

Once you have the z-score, you can use a z-table to find the probability of a value occurring within that standard deviation. This helps us understand what percentage of values fall below a particular point in our distribution.

The mean and variance are fundamental concepts in statistics that describe the central tendency and spread of a data set, respectively. The mean, often referred to as the average, is a measure that summarizes an entire dataset with a single value indicating the center of the data.

When it comes to variance, it's a measure of how much the values in a dataset differ from the mean. The standard deviation is simply the square root of the variance and provides a direct measure of how much the values deviate from the mean.

In the problem above, to find the mean \(\mu\) and variance \(\sigma^2\), we set up equations using z-scores. For example,

• Given \(Z = \frac{89 - \mu}{\sigma} = 1.28\), we can rearrange it to find \(\mu\), the mean.
• Similarly, \(Z = \frac{94 - \mu}{\sigma} = 1.64\) helps us establish another equation for \(\mu\) and \(\sigma\).

These allow us to solve for both the mean \(\mu\) and standard deviation \(\sigma\), and subsequently, the variance as it's \(\sigma^2\). Calculating these helps us describe the normal distribution fully with its parameters.

Probability is the measure of likelihood that a particular event will occur. It's the foundation upon which many statistical concepts are built, including the normal distribution, which shows how probabilities are distributed under a symmetric bell curve.

In the context of a normal distribution, probability tells us how likely it is for a value or a range of values to appear in our dataset. For example, in the exercise above, given probabilities are \(P(X < 89) = 0.90\) and \(P(X < 94) = 0.95\), which indicate that 90% and 95% of data fall below 89 and 94, respectively.

This is crucial because:

• It allows statisticians to make predictions about data.
• It helps in decision-making processes based on likely outcomes.
• It informs us about the relative standing of a value within a dataset.

Calculating z-scores and using them to explore probabilities in the context of the problem gives us an understanding of where data points lie within the distribution. Understanding this helps us grasp how data is likely to behave and allows us to make informed decisions or predictions based on statistical evidence.