Babies born weighing 2500 grams (about 5.5 pounds) or less are called low-birth-weight babies, and this condition sometimes indicates health problems for the infant. The mean birth weight for children born in a certain country is about 3494 grams (about 7.7 pounds). The mean birth weight for babies born one month early is 2632 grams. Suppose both standard deviations are 490 grams. Also assume that the distribution of birth weights is roughly unimodal and symmetric. Complete parts a through c below. a. Find the standardized score (z-score), relative to all births in the country, for a baby with a birth weight of 2500 grams. z = 2.03 (Round to two decimal places as needed.) b. Find the standardized score for a birth weight of 2500 grams for a child born one month early, using 2632 as the mean. Z= (Round to two decimal places as needed.)
The correct answer and explanation is:
Solution:
To calculate the z-score, we use the formula: z=x−μσz = \frac{x – \mu}{\sigma}
where:
- xx is the observed value,
- μ\mu is the mean,
- σ\sigma is the standard deviation.
Part (a): Z-Score Relative to All Births in the Country
For a baby with a birth weight of 2500 grams:
- x=2500x = 2500,
- μ=3494\mu = 3494 (mean for all births in the country),
- σ=490\sigma = 490.
Substitute the values into the formula: z=2500−3494490=−994490≈−2.03z = \frac{2500 – 3494}{490} = \frac{-994}{490} \approx -2.03
So, the z-score is: z=−2.03z = -2.03
Part (b): Z-Score Relative to Babies Born One Month Early
For a baby with a birth weight of 2500 grams:
- x=2500x = 2500,
- μ=2632\mu = 2632 (mean for one-month-early births),
- σ=490\sigma = 490.
Substitute the values into the formula: z=2500−2632490=−132490≈−0.27z = \frac{2500 – 2632}{490} = \frac{-132}{490} \approx -0.27
So, the z-score is: z=−0.27z = -0.27
Explanation:
The z-score represents how many standard deviations an observation is from the mean. A negative z-score indicates that the observation is below the mean, while a positive z-score indicates it is above the mean.
Part (a):
For the general population of all births, the z-score for a baby weighing 2500 grams is −2.03-2.03. This means that the baby’s weight is 2.03 standard deviations below the average weight of 3494 grams. A z-score of −2.03-2.03 is considered quite low, as it is rare for a birth weight to be this far below the mean. In a roughly normal distribution, only about 2.14% of babies have z-scores below −2-2, making this an uncommon case.
Part (b):
For babies born one month early, the z-score for a 2500-gram baby is −0.27-0.27. This z-score indicates that the baby’s weight is only 0.27 standard deviations below the mean weight of 2632 grams for early births. Since the z-score is close to zero, it suggests that the baby’s weight is relatively typical for a one-month-early birth. This weight is less unusual in this context compared to the general population.
By calculating these z-scores, we can contextualize the baby’s weight relative to different groups. While 2500 grams is significantly below average for the general population, it is much closer to the average for one-month-early births, illustrating the importance of comparing observations to the appropriate group.