To calculate the range of horn lengths for the middle 99.7% of Texas longhorn cattle, we can use the properties of a normal distribution, specifically the empirical rule (or 68-95-99.7 rule).
According to this rule:
- Approximately 68% of the data lies within one standard deviation of the mean.
- Approximately 95% of the data lies within two standard deviations of the mean.
- Approximately 99.7% of the data lies within three standard deviations of the mean.
Given:
- Mean (μ) = 60 inches
- Standard deviation (σ) = 4.5 inches
To find the range for the middle 99.7% of horn lengths:
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Calculate the lower bound: \[ \text{Lower bound} = \mu - 3\sigma = 60 - 3(4.5) = 60 - 13.5 = 46.5 , \text{inches} \]
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Calculate the upper bound: \[ \text{Upper bound} = \mu + 3\sigma = 60 + 3(4.5) = 60 + 13.5 = 73.5 , \text{inches} \]
Therefore, the range of horn lengths for the middle 99.7% of Texas longhorn cattle is 46.5 inches to 73.5 inches.
The correct response is: 46.5 in.–73.5 in.
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