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To determine if a value is significantly low or high, we use the concept of the z-score. A value is considered significantly low if its z-score is less than or equal to \textbf{-2}. Similarly, it is significantly high if its z-score is greater than or equal to \textbf{2}.
In the provided exercise, we are examining men's hip breadths to identify which are considered significantly low or high. The z-score helps standardize values, making it easier to identify outliers in a distribution.
The z-score is derived from the formula \text{ \[ z = \frac{x - \mu}{\sigma} \]}. By calculating this for the given mean and standard deviation, we can find the boundary values for significantly low and high hip breadths.

The \text{mean (\( \mu \))} and \text{standard deviation (\( \sigma \))} are fundamental concepts in statistics.

The \text{mean} represents the average of all values in a dataset. It is calculated by summing up all the values and dividing by the number of values. In this exercise, the mean hip breadth of men is given as \text{36.6 cm}.
The \text{standard deviation} measures the amount of dispersion or variation in a set of values. A low standard deviation indicates that the values are close to the mean, while a high standard deviation suggests that the values are spread out. Here, the standard deviation is \text{2.5 cm}.
Understanding the mean and standard deviation is crucial as they are used to calculate the z-score, helping to identify values that deviate significantly from the mean.

Calculating the boundary values involves using the z-score formula to find specific data points that mark the limits of being significantly low or high.

For significantly low values, we set the \text{z} to \text{-2} and solve using the formula:
\[ -2 = \frac{x - 36.6}{2.5} \]
Solving this equation, we get:
\[ -2 \times 2.5 = x - 36.6 \]\[ -5 = x - 36.6 \]\[ x = 31.6 \]Any hip breadth less than or equal to \text{31.6 cm} is significantly low.

For significantly high values, we set the \text{z} to \text{2} and use the same z-score formula:
\[ 2 = \frac{x - 36.6}{2.5} \]Solving this, we find:
\[ 2 \times 2.5 = x - 36.6 \]\[ 5 = x - 36.6 \]\[ x = 41.6 \]So, any hip breadth greater than or equal to \text{41.6 cm} is significantly high.
By calculating these boundary values, we can easily identify the hip breadths that fall outside the typical range.