To find the baby kitten's weight [tex]\( X \)[/tex] that corresponds to a [tex]\( Z \)[/tex]-score of 1.49, we can use the formula for the [tex]\( Z \)[/tex]-score:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
where:
- [tex]\( Z \)[/tex] is the [tex]\( Z \)[/tex]-score,
- [tex]\( \mu \)[/tex] is the mean of the dataset,
- [tex]\( \sigma \)[/tex] is the standard deviation, and
- [tex]\( X \)[/tex] is the value we want to find.
We are given the values:
- [tex]\( Z = 1.49 \)[/tex]
- [tex]\( \mu = 10.2 \)[/tex] (mean)
- [tex]\( \sigma = 1.5 \)[/tex] (standard deviation)
We need to rearrange the formula to solve for [tex]\( X \)[/tex]. Start by multiplying both sides by [tex]\( \sigma \)[/tex]:
[tex]\[ Z \cdot \sigma = X - \mu \][/tex]
Next, add [tex]\( \mu \)[/tex] to both sides to isolate [tex]\( X \)[/tex]:
[tex]\[ Z \cdot \sigma + \mu = X \][/tex]
Substitute the given values into the equation:
[tex]\[ 1.49 \cdot 1.5 + 10.2 = X \][/tex]
Perform the multiplication and addition:
[tex]\[ 2.235 + 10.2 = X \][/tex]
[tex]\[ X = 12.435 \][/tex]
Therefore, the baby kitten's weight that corresponds to a [tex]\( Z \)[/tex]-score of 1.49 is approximately [tex]\( 12.435 \)[/tex].
