To find the z-score of a newborn who weighs 4,000 grams, we need to use the z-score formula. The z-score tells us how many standard deviations a given value is from the mean. The formula for calculating the z-score is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where
- [tex]\( X \)[/tex] is the value you are analyzing (in this case, the newborn's weight, which is 4,000 grams),
- [tex]\( \mu \)[/tex] is the mean of the population, and
- [tex]\( \sigma \)[/tex] is the standard deviation of the population.
For this problem, assume the following statistics about the population of newborns:
- The mean weight of newborns ([tex]\( \mu \)[/tex]) is 3,500 grams.
- The standard deviation ([tex]\( \sigma \)[/tex]) is 500 grams.
Now, we substitute these values into the z-score formula:
[tex]\[
z = \frac{4,000 - 3,500}{500}
\][/tex]
First, calculate the numerator (the difference between the newborn's weight and the mean):
[tex]\[
4,000 - 3,500 = 500
\][/tex]
Next, divide this result by the standard deviation (500 grams):
[tex]\[
z = \frac{500}{500} = 1.0
\][/tex]
Thus, the z-score for a newborn who weighs 4,000 grams is [tex]\( 1.0 \)[/tex]. This means that the newborn's weight is 1 standard deviation above the mean weight of newborns in this population.
