Question 1191332: The distribution of the weights of a sample of 1,400 cargo containers is symmetric and bell-shaped
According to the Empirical Rule, what percent of the weights will lie between X⎯⎯⎯−2s and X⎯⎯⎯+2s ?
According to the Empirical Rule, what percent of the weights will lie between X⎯⎯⎯ and X⎯⎯⎯+2s ?
According to the Empirical Rule, what percent of the weights will lie below X⎯⎯⎯−2s ?
i need to see steps
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how the Empirical Rule applies to this situation:
* **Empirical Rule:** The Empirical Rule (also known as the 68-95-99.7 Rule) applies to distributions that are symmetric and bell-shaped (approximately normal). It states that:
* Approximately 68% of the data falls within 1 standard deviation of the mean (between X̄ - s and X̄ + s).
* Approximately 95% of the data falls within 2 standard deviations of the mean (between X̄ - 2s and X̄ + 2s).
* Approximately 99.7% of the data falls within 3 standard deviations of the mean (between X̄ - 3s and X̄ + 3s).
* **Between X̄ - 2s and X̄ + 2s:** According to the Empirical Rule, approximately **95%** of the weights will lie between X̄ - 2s and X̄ + 2s.
* **Between X̄ and X̄ + 2s:** Since the distribution is symmetric, half of the 95% will lie between the mean and 2 standard deviations above the mean. Therefore, approximately 95%/2 = **47.5%** of the weights will lie between X̄ and X̄ + 2s.
* **Below X̄ - 2s:** Since 95% of the data is between X̄ - 2s and X̄ + 2s, the remaining 5% is outside this range. Because the distribution is symmetric, half of this 5% (2.5%) will be below X̄ - 2s, and the other half (2.5%) will be above X̄ + 2s. Therefore, approximately **2.5%** of the weights will lie below X̄ - 2s.
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