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.
To find the range of horn lengths for the middle 95% of Texas Longhorn cattle, we need to calculate the two values that represent the boundaries of the middle 95% of the distribution.
Step 1: Find the z-score for the 95th percentile.
To determine this, we need to find the z-score associated with the cumulative probability of 0.95. We can use the z-table or a calculator to find that the z-score for a cumulative probability of 0.95 is approximately 1.96.
Step 2: Use the z-score to find the corresponding horn length.
We can use the formula z = (X - μ) / σ, where X is the horn length, μ is the mean horn length, and σ is the standard deviation. Rearranging the formula to solve for X, we have X = (z * σ) + μ.
In this case, we want to find the horn length associated with a z-score of 1.96. Plugging in the values, we get X = (1.96 * 4.5) + 60 = 68.82 inches.
Step 3: Calculate the lower boundary of the middle 95% range.
Since the distribution is symmetric, the lower boundary will be the negative of the upper boundary. Therefore, the lower boundary is 60 - (68.82 - 60) = 51.18 inches.
Step 4: Calculate the upper boundary of the middle 95% range.
The upper boundary is simply the horn length associated with the z-score of 1.96: 68.82 inches.
Therefore, the range of horn lengths for the middle 95% of Texas Longhorn cattle is 51.18 to 68.82 inches.