Final answer:
To find the number of newborn boys expected to be between 45.2 and 54.8 cm tall, we apply the empirical rule to the normal distribution with a mean of 50 cm and a standard deviation of 1.58 cm. Approximately 99.7% are expected within 3 standard deviations, which corresponds to 997 of the 1000 boys, but the closest option is D. 990.
Step-by-step explanation:
The question involves the application of the normal distribution and the empirical rule to determine how many newborn boys fall within a certain range of heights. Given that the distribution of height among newborn boys has a mean (μ) of 50 cm and a variance (σ²) of 2.5 cm², we can infer that the standard deviation (σ) is the square root of the variance, which equals 1.58 cm. Using the empirical rule:
- Approximately 68% of the data falls within 1 σ of the mean (50 cm ± 1.58 cm).
- Approximately 95% falls within 2 σ of the mean (50 cm ± 3.16 cm).
- Approximately 99.7% falls within 3 σ of the mean (50 cm ± 4.74 cm).
The range from 45.2 to 54.8 cm is approximately 3 σ from the mean, which captures about 99.7% of the data according to the empirical rule. Therefore, of the 1000 newborn boys, we would expect approximately 99.7% of them to fall within this range. This corresponds to 997 boys, but since we are limited to the options provided, the closest answer is D. 990.