To find the probability that a randomly selected medical student had a total score less than 490, we need to calculate the z-score first and then use the standard normal distribution.

Step 1: Calculate the z-score

The z-score formula is given by:

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

Where:

• $X = 490$ (the score we are comparing),
• $\mu = 500$ (the mean score),
• $\sigma = 10.7$ (the standard deviation).

Substituting the values:

$z = \frac{490 - 500}{10.7} = \frac{-10}{10.7} \approx -0.9346$

Step 2: Find the probability

The probability corresponds to the area to the left of the z-score. We can look up the z-score of $-0.9346$ in the standard normal distribution table or use a calculator or software to find the corresponding probability.

Using a standard normal distribution table or a calculator:

$P(Z < -0.9346) \approx 0.1747$

Thus, the probability that a randomly selected student scored less than 490 is approximately 0.1747.

Final Answer:

The probability is 0.1747 (rounded to four decimal places).

Would you like more details or have additional questions on this? Here are some related questions:

1. How do you find the probability for scores greater than a given value in a normal distribution?
2. How do you interpret a z-score and its associated probability in real-world terms?
3. What is the impact of changing the mean or standard deviation on the probability?
4. How can you find the probability of a score falling between two values in a normal distribution?
5. How does the normal distribution relate to other types of probability distributions?

Tip: To calculate probabilities for other values, simply calculate the corresponding z-scores and refer to the standard normal distribution table or use a calculator with statistical functions.