Question

The mean weight of an elephant is 244 pounds with a standard deviation of 15 pounds. A herd of elephants were spotted in Africa. An elephant will be randomly selected from the herd. What is the probability that a randomly selected elephant will weigh between 229 and 289?

Answer 1

To calculate the probability that a randomly selected elephant will weigh between 229 and 289 pounds, we will use the properties of the normal distribution.

Given:
- Mean (\(\mu\)) = 244 pounds
- Standard deviation (\(\sigma\)) = 15 pounds

We will standardize the weights using the z-score formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]

1. **Calculate the z-scores for 229 pounds and 289 pounds.**

For \(X = 229\):
\[
z_{229} = \frac{(229 - 244)}{15} = \frac{-15}{15} = -1
\]

For \(X = 289\):
\[
z_{289} = \frac{(289 - 244)}{15} = \frac{45}{15} = 3
\]

2. **Use the z-table (or standard normal distribution table) to find the probabilities corresponding to these z-scores.**

- For \(z = -1\):
The probability \(P(Z < -1)\) is approximately 0.1587.

- For \(z = 3\):
The probability \(P(Z < 3)\) is approximately 0.9987.

3. **Now, find the probability that a randomly selected elephant weighs between 229 and 289 pounds:**
\[
P(229 < X < 289) = P(Z < 3) - P(Z < -1)
\]
\[
P(229 < X < 289) = 0.9987 - 0.1587 = 0.8400
\]

Thus, the probability that a randomly selected elephant will weigh between 229 and 289 pounds is approximately **0.8400** or **84.00%**.