To solve $P(184.5 < X < 237.4)$, we need to standardize the values using the Z-score formula:

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

Where:

• $\mu = 33.5$ (mean)
• $\sigma = 75.5$ (standard deviation)
• $X$ is the value we're calculating the Z-score for.

Step 1: Compute Z-scores

For $X = 184.5$:

$Z_1 = \frac{184.5 - 33.5}{75.5} = \frac{151.0}{75.5} \approx 2.000$

For $X = 237.4$:

$Z_2 = \frac{237.4 - 33.5}{75.5} = \frac{203.9}{75.5} \approx 2.701$

Step 2: Find the probabilities using the standard normal distribution

Using a Z-table or a calculator:

• $P(Z_1 = 2.000) \approx 0.9772$
• $P(Z_2 = 2.701) \approx 0.9964$

Step 3: Calculate the probability

The probability that $X$ is between 184.5 and 237.4 is:

$P(184.5 < X < 237.4) = P(Z_2) - P(Z_1)$

$P(184.5 < X < 237.4) = 0.9964 - 0.9772 = 0.0192$

Final Answer:

$P(184.5 < X < 237.4) = 0.0192$