Question

Suppose X ~ N( 13.5, 1.5), and x = 9. Find and interpret the z-score of the standardized normal random variable.

Answer 1

The z-score for x = 9 when X ~ N(13.5, 1.5) is -3, indicating that x is three standard deviations below the mean of this normal distribution.

Step-by-step explanation:

To find the z-score of the standardized normal random variable when given X ~ N(13.5, 1.5), and x = 9, we utilize the z-score formula, which is z = (x - μ) / σ. The mean (μ) is 13.5 and the standard deviation (σ) is 1.5. To calculate:

z = (9 - 13.5) / 1.5

z = -4.5 / 1.5

z = -3

This z-score indicates that the value x = 9 is three standard deviations below the mean of the normal distribution X. To interpret this, a z-score of -3 is quite far from the mean on the left side of the normal curve, showing that x = 9 is an uncommon or extreme value in this distribution.