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
[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
