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A z-score is a valuable statistical tool that helps us understand where a particular data point stands in a normal distribution. Essentially, it represents the number of standard deviations a data point is from the mean. Calculating the z-score is simple with the formula:

• \( Z = \frac{X - \mu}{\sigma} \)

Here, \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. For instance, a z-score of 0 indicates the data point is exactly at the mean. A positive z-score means it's above the mean, while a negative z-score means it's below. For example, the z-score for a family spending \(430 is calculated as follows:

• \( Z = \frac{430 - 490}{90} = -0.67 \)

This implies that \)430 is 0.67 standard deviations below the mean of $490. Understanding z-scores allows for determining the probability of a value lying within a specific range.

Standard deviation is a key concept in statistics, providing a measure of the dispersion or spread of a set of data values around their mean. In our context, the standard deviation is $90, which means that a typical family's food expenditure is likely to vary by $90 above or below the mean of $490.
Here's a breakdown:

• Low Standard Deviation: Indicates that the data points are closely clustered around the mean.
• High Standard Deviation: Suggests that the data points are spread out over a larger range of values.

Being aware of the standard deviation enables us to determine how typical or atypical certain data points are within the context of the entire dataset. For example, if a family's expenditure is $600, they are further from the average, as indicated by their relatively high z-score.

Probability tells us how likely an event is to occur. In the context of normal distribution, we can calculate the probability of a data point falling within a certain range using a z-score table or normal distribution calculator. It's expressed as a percentage but can also be a fraction or decimal. For example, to find the probability of a family's expenses being between \(430 and \)600, we look at their respective z-scores:

• Z-score for \(430: \( -0.67 \)
• Z-score for \)600: \( 1.22 \)

Using a z-table, we find:

• Probability that expenses are less than \(430 is approximately 25.14%.
• Probability that expenses are less than \)600 is approximately 88.88%.

The difference, 63.74%, shows the probability of a family's monthly spending falling between these values. Probability is essential in understanding and interpreting statistical data, especially in real-world scenarios like budgeting for a family's monthly expenses.

In statistics, the mean is a measure of central tendency and is simply the average of a set of numbers. In the problem here, the mean food expenditure for a family of four is $490.
This mean is indicative of what is typical for this group:

• Calculate the mean by adding all the observed data points and dividing by the number of points.
• The mean represents the central point of a data set.

In the context of a normal distribution, the mean is the peak of the bell curve. It helps in understanding where most families' expenditures lie. For this exercise, having a mean of $490 means this is the central value around which the family expenditures are distributed. Knowing the mean enables easier calculation and understanding of other statistical measures like standard deviation and z-scores.