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Normal Distribution Height

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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%**.