Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?

Verified step by step guidance

1. Identify the random variable \( X \) representing the annual rainfall, which follows a normal distribution with mean \( \mu = 20.11 \) inches and standard deviation \( \sigma = 4.7 \) inches.
2. Standardize the value 17 inches to find the corresponding z-score using the formula: \[ Z = \frac{X - \mu}{\sigma} = \frac{17 - 20.11}{4.7} \]
3. Calculate the z-score to determine how many standard deviations 17 inches is below the mean rainfall.
4. Use the standard normal distribution table (or a calculator) to find the probability \( P(Z \leq z) \) corresponding to the calculated z-score.
5. Since the problem asks for the probability that rainfall exceeds 17 inches, compute \( P(X > 17) = 1 - P(Z \leq z) \).

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve defined by its mean and standard deviation. It models many natural phenomena, including rainfall amounts, allowing us to calculate probabilities for ranges of values.

Standardization (Z-Score)

Standardization converts a normal random variable to a standard normal variable with mean 0 and standard deviation 1 using the formula Z = (X - μ) / σ. This transformation enables the use of standard normal tables or software to find probabilities.

Probability Calculation for Normal Variables

To find the probability that a normal variable exceeds a certain value, we calculate the corresponding Z-score and then use the standard normal distribution to find the area to the right of that Z. This area represents the probability of exceeding the given value.