To determine the required values, we will use the information provided about the mean ([tex]\(\mu\)[/tex]) and the standard deviation ([tex]\(\sigma\)[/tex]) of birth weights of newborn babies.
Given:
- Mean ([tex]\(\mu\)[/tex]) = 3500 g
- Standard Deviation ([tex]\(\sigma\)[/tex]) = 500 g
We will use the z-score formula:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
### 1. Z-score for a newborn weighing 2500 g
To find the z-score for a newborn weighing 2500 g:
[tex]\[ x = 2500 \, \text{g} \][/tex]
[tex]\[ z = \frac{2500 - 3500}{500} = \frac{-1000}{500} = -2.0 \][/tex]
So, the z-score for a 2500 g baby is [tex]\(\boxed{-2.0}\)[/tex].
### 2. Z-score for a newborn weighing 4000 g
To find the z-score for a newborn weighing 4000 g:
[tex]\[ x = 4000 \, \text{g} \][/tex]
[tex]\[ z = \frac{4000 - 3500}{500} = \frac{500}{500} = 1.0 \][/tex]
So, the z-score for a 4000 g baby is [tex]\(\boxed{1.0}\)[/tex].
### 3. Weight that would give a newborn a z-score of -0.75
We are given the z-score and need to find the corresponding weight. We rearrange the z-score formula to solve for [tex]\(x\)[/tex]:
[tex]\[ z = -0.75 \][/tex]
[tex]\[ x = z \cdot \sigma + \mu \][/tex]
Substituting the values in:
[tex]\[ x = -0.75 \cdot 500 + 3500 \][/tex]
[tex]\[ x = -375 + 3500 \][/tex]
[tex]\[ x = 3125 \, \text{g} \][/tex]
So, the weight that would give a newborn a z-score of -0.75 is [tex]\(\boxed{3125}\)[/tex] grams.
