Sure, let's go through this step-by-step.
First, we need to understand the problem. We have a normally distributed variable [tex]\( X \)[/tex] with a mean ([tex]\(\mu\)[/tex]) of 9 and a standard deviation ([tex]\(\sigma\)[/tex]) of 1.5. We are given a specific value, [tex]\( x = 13.5 \)[/tex], and we need to find the corresponding [tex]\( z \)[/tex]-score.
Step 1: Recall the formula for the [tex]\( z \)[/tex]-score.
The [tex]\( z \)[/tex]-score for a value [tex]\( x \)[/tex] in a normal distribution is calculated by:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
Step 2: Substitute the given values into the formula.
We know:
- [tex]\( x = 13.5 \)[/tex]
- [tex]\( \mu = 9 \)[/tex]
- [tex]\( \sigma = 1.5 \)[/tex]
So,
[tex]\[ z = \frac{13.5 - 9}{1.5} \][/tex]
Step 3: Perform the arithmetic operation.
[tex]\[ z = \frac{4.5}{1.5} \][/tex]
[tex]\[ z = 3 \][/tex]
Step 4: Interpret the [tex]\( z \)[/tex]-score.
- The [tex]\( z \)[/tex]-score is 3.0.
- This means that the value [tex]\( x = 13.5 \)[/tex] is 3 standard deviations to the right of the mean.
Now we can fill in the blanks in the given interpretation:
The [tex]\( z \)[/tex]-score when [tex]\( x = 13.5 \)[/tex] is [tex]\( 3.0 \)[/tex]. The mean is [tex]\( 9 \)[/tex].
This [tex]\( z \)[/tex]-score tells you that [tex]\( x = 13.5 \)[/tex] is [tex]\( 3.0 \)[/tex] standard deviations to the right of the mean.
