Sure! Let's solve this problem step by step:
1. Identify the Mean and Standard Deviation:
The mean (\(\mu\)) of the birth weights is 3,500 grams.
The standard deviation (\(\sigma\)) of the birth weights is 500 grams.
2. Understanding the Empirical Rule:
The empirical rule states that for a normally distributed dataset:
- Approximately 68% of the data falls within one standard deviation (\(\sigma\)) of the mean (\(\mu\)).
3. Calculate the Range Using the Empirical Rule:
To find the range that contains 68% of the data, you need to calculate:
- The lower bound of the range, which is \(\mu - \sigma\).
- The upper bound of the range, which is \(\mu + \sigma\).
4. Perform the Calculations:
- The lower bound is \(3,500 \, \text{g} - 500 \, \text{g} = 3,000 \, \text{g}\).
- The upper bound is \(3,500 \, \text{g} + 500 \, \text{g} = 4,000 \, \text{g}\).
Therefore, according to the empirical rule, 68% of all newborn babies in the United States weigh between [tex]\(3,000 \, \text{g}\)[/tex] and [tex]\(4,000 \, \text{g}\)[/tex].
