Let's find the [tex]\( z \)[/tex]-score for a newborn who weighs [tex]\( 4,000 \)[/tex] grams. We're given the mean ([tex]\(\mu\)[/tex]) and the standard deviation ([tex]\(\sigma\)[/tex]) of the birth weights. The mean weight of newborn babies is [tex]\(\mu = 3,500 \)[/tex] grams, and the standard deviation is [tex]\(\sigma = 500 \)[/tex] grams.
We use the formula for the [tex]\( z \)[/tex]-score:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
where:
- [tex]\( x \)[/tex] is the observed value (the weight of the newborn in this case),
- [tex]\(\mu\)[/tex] is the mean,
- [tex]\(\sigma\)[/tex] is the standard deviation.
For a newborn weighing [tex]\( 4,000 \)[/tex] grams, we substitute the values into the formula:
[tex]\[
z = \frac{4,000 - 3,500}{500}
\][/tex]
First, we calculate the difference between the weight of the newborn ([tex]\( x \)[/tex]) and the mean ([tex]\(\mu\)[/tex]):
[tex]\[
4,000 - 3,500 = 500
\][/tex]
Next, we divide this difference by the standard deviation ([tex]\(\sigma\)[/tex]):
[tex]\[
z = \frac{500}{500} = 1.0
\][/tex]
Therefore, the [tex]\( z \)[/tex]-score of a newborn who weighs [tex]\( 4,000 \)[/tex] grams is [tex]\( 1.0 \)[/tex].
