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 99.7% of Texas longhorn cattle.
a. 51 in. - 60 in.
b. 51 in. - 69 in.
c. 60 in. - 73.5 in.
d. 46.5 in. - 73.5 in.
To calculate the range of horn lengths for the middle 99.7% of Texas longhorn cattle, we need to find the z-scores corresponding to the upper and lower tails of the distribution.
First, we find the z-score corresponding to the upper tail of 0.15% (0.3%/2 = 0.15%). Using a standard normal distribution table or calculator, we find this z-score to be approximately 3.17.
Next, we can use the formula z = (x - μ) / σ to find the corresponding value for x, the horn length.
For the upper tail:
3.17 = (x - 60) / 4.5
Solving for x:
3.17 * 4.5 + 60 = x
14.265 + 60 = x
74.265 = x
For the lower tail, the z-score is -3.17.
-3.17 = (x - 60) / 4.5
Solving for x:
-3.17 * 4.5 + 60 = x
-14.265 + 60 = x
45.735 = x
Therefore, the range of horn lengths for the middle 99.7% of Texas longhorn cattle is approximately 45.735 inches to 74.265 inches.
The correct answer is: d. 46.5 in. - 73.5 in.