Normal Distribution Height
1. **State the problem:** We have a normally distributed variable representing the height of dwarf cocoa plants with mean $\mu = 120$ cm and standard deviation $\sigma = 8$ cm. We want to find the probability that a randomly selected plant is taller than 130 cm.
2. **Convert the height to a standard normal variable:** We use the formula for the z-score:
$$ z = \frac{X - \mu}{\sigma} $$
where $X = 130$ cm.
3. **Calculate the z-score:**
$$ z = \frac{130 - 120}{8} = \frac{10}{8} = 1.25 $$
4. **Find the probability:** We want $P(X > 130) = P(Z > 1.25)$ where $Z$ is a standard normal variable.
5. **Use standard normal distribution tables or a calculator:**
$P(Z > 1.25) = 1 - P(Z \leq 1.25)$.
From the standard normal table, $P(Z \leq 1.25) \approx 0.8944$.
6. **Calculate the final probability:**
$$ P(Z > 1.25) = 1 - 0.8944 = 0.1056 $$
**Answer:** The probability that a randomly selected dwarf cocoa plant is taller than 130 cm is approximately **0.1056** or **10.56%**.