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. Use the Empirical Rule.
The Empirical Rule states that for a normally distributed data set, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations of the mean, and approximately 99.7% falls within three standard deviations of the mean.
In this case, we want to find the range of horn lengths for the middle 95% of Texas longhorn cattle. Since 95% falls within two standard deviations of the mean, we need to find the values that are two standard deviations below and two standard deviations above the mean.
Two standard deviations below the mean would be 60 - (2 * 4.5) = 60 - 9 = 51 inches.
Two standard deviations above the mean would be 60 + (2 * 4.5) = 60 + 9 = 69 inches.
Therefore, the range of horn lengths for the middle 95% of Texas longhorn cattle is 51 inches to 69 inches.