College

5) A population distribution has a standard deviation of 5. What position in this distribution is identified by a z-score of -2.00?

A. 10 points below the mean
B. 2 points above the mean
C. 2 points below the mean
D. 10 points above the mean

Answer :

To find the specific position in a population distribution that corresponds to a given z-score, you can use the z-score formula. The z-score formula is:

[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]

where:
- [tex]\( z \)[/tex] is the z-score,
- [tex]\( X \)[/tex] is the value in the distribution,
- [tex]\( \mu \)[/tex] is the mean of the distribution,
- [tex]\( \sigma \)[/tex] is the standard deviation.

We can rearrange this formula to solve for [tex]\( X \)[/tex], the value in the distribution:

[tex]\[ X = z \cdot \sigma + \mu \][/tex]

In this question:
- The standard deviation [tex]\( \sigma \)[/tex] is given as 5.
- The z-score [tex]\( z \)[/tex] is given as -2.00.
- We need to find the position [tex]\( X \)[/tex].

Let's plug the values into our rearranged formula:

[tex]\[ X = (-2.00) \cdot 5 + \mu \][/tex]

Simplify the multiplication:

[tex]\[ X = -10 + \mu \][/tex]

This tells us that the position in the distribution is 10 points below the mean.

Thus, the correct answer is:
- 10 points below the mean