Questions: During monsoon season in India, the total amount of rainfall follows a normal distribution with a mean of 88 centimeters and a standard deviation of 5.2 centimeters. What is the probability that a monsoon season will have a total rainfall greater than 93 centimeters? Type your answer as a decimal with four decimal places.

Solution Steps

We are tasked with finding the probability that the total rainfall during the monsoon season in India exceeds \( 93 \) centimeters. The rainfall follows a normal distribution with a mean \( \mu = 88 \) cm and a standard deviation \( \sigma = 5.2 \) cm.

To find the probability, we first calculate the Z-score for \( 93 \) cm using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

Substituting the values:

\[ Z_{end} = \frac{93 - 88}{5.2} = \frac{5}{5.2} \approx 0.9615 \]

Next, we need to find the cumulative probability \( P(X < 93) \), which is represented as \( \Phi(Z_{end}) \). From the output, we have:

\[ P(X < 93) = \Phi(0.9615) \approx 0.8319 \]

To find the probability that the rainfall exceeds \( 93 \) cm, we calculate the complement:

\[ P(X > 93) = 1 - P(X < 93) = 1 - 0.8319 \approx 0.1681 \]

Final Answer