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The Z-score is a crucial concept in statistics, particularly when dealing with normal distributions. It tells us how far away a particular data point is from the mean, in terms of standard deviations. If you imagine the normal distribution curve as a mountain, the Z-score helps you locate any specific point on this mountain.

Z-scores are calculated using the formula:

• \( Z = \frac{X - \mu}{\sigma} \)

In this formula, \(X\) is our data point, \(\mu\) represents the mean of the distribution, and \(\sigma\) is the standard deviation.

These scores are valuable because they allow us to compare different data sets by standardizing scores, even if the datasets have different means and standard deviations. For example, a Z-score of 1 means the value is one standard deviation above the mean, whereas a Z-score of -1.28 like in our exercise indicates the value is just over one standard deviation below the mean.

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. It is often denoted by \(\sigma\). A low standard deviation indicates that the data points are closely clustered around the mean, whereas a high standard deviation indicates that the data points are more spread out.

Understanding this concept is essential because it helps us gauge the consistency of the data.

In a normal distribution, standard deviation determines the shape of the curve:

• A smaller standard deviation results in a steeper, narrower curve.
• A larger standard deviation leads to a flatter, wider curve.

For instance, in our exercise, we had two different normal distributions: \(N(5,2)\) and \(N(500,25)\). The standard deviations (2 and 25, respectively) tell us how "spread out" the data is from the means 5 and 500. Understanding standard deviation helps us interpret the distribution's spread and variability.

Cumulative probability is the likelihood that a randomly selected value from a probability distribution will be less than or equal to a specified value. It is a crucial element when working with normal distribution as it helps in finding the probability of a data point being within a certain range.

For any given Z-score, the cumulative probability can be found using Z-tables or calculators which provide the area under the curve to the left of the Z-score.

An understanding of cumulative probability is essential for interpreting our original exercise. In part (a), a cumulative probability of 0.10 indicates that approximately 10% of values fall below the endpoint, corresponding to a Z-score of about -1.28. In part (b), since the area to the right was given, we subtracted the probability from 1 to find the left area or cumulative probability, resulting in 0.95 and a Z-score of approximately 1.645.

The concept allows us to assess the probability of variables falling within specific ranges on a normal distribution, which is fundamental for statistical analysis and decision-making.