Explain why P(X < 30) should be reported as < 0.0001 if X is a normal random variable with mean 100 and standard deviation 15.

Verified step by step guidance

1. Identify the distribution of the random variable \(X\). Here, \(X\) is normally distributed with mean \(\mu = 100\) and standard deviation \(\sigma = 15\).
2. Convert the value 30 to a standard normal variable \(Z\) using the formula \(Z = \frac{X - \mu}{\sigma}\). So, calculate \(Z = \frac{30 - 100}{15}\).
3. Interpret the \(Z\)-score: since 30 is far below the mean of 100, the \(Z\)-score will be a large negative number, indicating a value far in the left tail of the normal distribution.
4. Use standard normal distribution tables or software to find \(P(Z < z)\) for the calculated \(Z\)-score. Because the \(Z\)-score is very low, this probability will be extremely small.
5. Since probabilities less than 0.0001 are often reported as \(< 0.0001\) due to rounding and practical significance, \(P(X < 30)\) should be reported as \(< 0.0001\) to reflect the rarity of such an event.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric bell-shaped curve, defined by its mean and standard deviation. It models many natural phenomena and allows calculation of probabilities for ranges of values.

Standardization (Z-Score)

Standardization converts a normal random variable to a standard normal variable by subtracting the mean and dividing by the standard deviation. This Z-score allows use of standard normal tables to find probabilities associated with any value.

Tail Probability and Reporting Small Probabilities

When a value lies far in the tail of the distribution, the probability of observing such an extreme value is very small. Probabilities less than 0.0001 are often reported as '< 0.0001' to indicate extreme rarity and limitations in numerical precision.