91Ó°ÊÓ

When discussing statistics, two fundamental concepts are the mean and standard deviation.
The mean, denoted as \( \mu \ \), is the average value of a dataset, calculated by summing all observations and dividing by the number of observations. In our exercise, the mean hip breadth for men is 36.6 cm.
The standard deviation, represented by \( \sigma \ \), measures the spread or dispersion of the dataset around the mean. A small standard deviation indicates that the data points are close to the mean, while a large standard deviation shows that the data points are spread out over a wider range. Here, the standard deviation of men's hip breadths is 2.5 cm.
Understanding these two metrics helps in identifying how individual data points deviate from the average, which is crucial for computing z-scores.

A value is considered significantly low if it is at or below a z-score of -2. The z-score indicates how many standard deviations a data point is from the mean. To determine a significantly low hip breadth, we use the z-score formula: \[ Z = \frac{X - \mu}{\sigma} \]. Set \ Z = -2 \ : \[ -2 = \frac{X - 36.6}{2.5} \] Solving for X (the significantly low value), we get:

• Multiply both sides by 2.5: \ (-2) \times \ 2.5 = \ X - 36.6 \ \[ -5 = \ X - 36.6 \].
• Add 36.6 to both sides: \[ X = 31.6 \CM \].

Therefore, any hip breadth less than or equal to 31.6 cm is significantly low.

Similarly, a value is significantly high if it has a z-score at or above 2. This means it's two standard deviations above the mean. Using the z-score formula, we set \ Z = 2:
\[ 2 = \frac{X - 36.6}{2.5} \]
Following the same steps to solve for X:

• Multiply both sides by 2.5: \[ 2 \times \ 2.5 = \ X - 36.6 \] \ \[ 5 = X - 36.6 \]
• Add 36.6 to both sides: \[ X = 41.6 \CM \].

Thus, a hip breadth greater than or equal to 41.6 cm is classified as significantly high.