Horn lengths of Texas longhorn cattle are normally distributed. The mean horn spread is 60 inches with a standard deviation of 4.5 inches. Calculate the range of horn lengths for the middle 95% of Texas Longhorn cattle. Show your work or explain how you got your answer.

Answer 1

To calculate the range of horn lengths for the middle 95% of Texas Longhorn cattle, we will use the properties of the normal distribution.

1. Identify the mean and standard deviation:

   • Mean (\(\mu\)) = 60 inches
   • Standard deviation (\(\sigma\)) = 4.5 inches

2. Determining the z-scores for the middle 95%: The middle 95% of a normal distribution corresponds to the range between the 2.5th percentile and the 97.5th percentile. Looking up these percentiles in the standard normal distribution table, we find that:

   • The z-score for the 2.5th percentile is approximately \(-1.96\).
   • The z-score for the 97.5th percentile is approximately \(1.96\).

3. Using the z-scores to find the horn lengths: We can convert these z-scores back to the original scale of horn lengths using the formula:

   \[ X = \mu + z \cdot \sigma \]

   • For the 2.5th percentile: \[ X_{2.5} = 60 + (-1.96) \cdot 4.5 \] \[ X_{2.5} = 60 - 8.82 = 51.18 \text{ inches} \]

   • For the 97.5th percentile: \[ X_{97.5} = 60 + 1.96 \cdot 4.5 \] \[ X_{97.5} = 60 + 8.82 = 68.82 \text{ inches} \]

4. Conclusion: Thus, the range of horn lengths for the middle 95% of Texas Longhorn cattle is from approximately 51.18 inches to 68.82 inches.