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 long horn cattle. show your work or explain how you got your answer
To find the range of horn lengths for the middle 95% of Texas longhorn cattle, we need to determine the z-values corresponding to the lower and upper boundaries of this range.
Step 1: Find the z-value corresponding to the lower boundary:
Since we want to calculate the range for the middle 95%, we need to find the z-value that cuts off 2.5% from the tail on the left side of the distribution.
Using a standard normal distribution table or a z-table calculator, we find that the z-value corresponding to the 2.5th percentile (0.025) is approximately -1.96.
Step 2: Find the z-value corresponding to the upper boundary:
Similarly, we need to find the z-value that cuts off 2.5% from the tail on the right side of the distribution.
Using the same reasoning as in step 1, we find that the z-value corresponding to the 97.5th percentile (1 - 0.025) is approximately 1.96.
Step 3: Calculate the horn lengths for the lower and upper boundaries:
The z-value formula is: z = (x - mean) / standard deviation.
Rearranging the formula, we have: x = z * standard deviation + mean.
For the lower boundary:
x = -1.96 * 4.5 + 60 = 51.18 inches.
For the upper boundary:
x = 1.96 * 4.5 + 60 = 68.22 inches.
Therefore, the range of horn lengths for the middle 95% of Texas longhorn cattle is approximately 51.18 inches to 68.22 inches.