To compute the probability $P(X < 100)$ for a normally distributed population with a mean $\mu = 120$ and standard deviation $\sigma = 20$, we will standardize the variable using the z-score formula:

$Z = \frac{X - \mu}{\sigma}$

Here:

• $X = 100$
• $\mu = 120$
• $\sigma = 20$

Now, substitute the values into the formula:

$Z = \frac{100 - 120}{20} = \frac{-20}{20} = -1$

Next, we need to find the probability corresponding to $Z < -1$ using the standard normal distribution table or a calculator.

The probability $P(Z < -1)$ corresponds to approximately 0.1587.

Thus, $P(X < 100) \approx 0.1587$.

Summary:

The probability that $X$ is less than 100 is approximately 0.1587, or 15.87%.

Would you like more details or further explanation?

Related Questions:

1. What is $P(X > 140)$ for the same distribution?
2. How do you compute $P(100 < X < 140)$?
3. How do you standardize a normal distribution?
4. What is the cumulative probability for $Z = 0$?
5. How do you interpret the z-score of a data point?

Tip: To find probabilities in a normal distribution, always standardize the variable first using the z-score formula.