Final answer:
To find the probabilities, we use the standardization formula and the standard normal distribution table. For P(X > 13.5), we standardize 13.5 and find the corresponding probability. For P(X < 9.4), we standardize 9.4 and find its probability.
Step-by-step explanation:
To find the probability that a normal random variable X is greater than a specific value, we can use the standardization formula and the standard normal distribution table.
For part (a), we want to find P(X > 13.5). First, we standardize 13.5 using the formula z = (x - μ) / σ, where x = 13.5, μ = 10, and σ = 2. Plugging these values into the formula, we get z = (13.5 - 10) / 2 = 1.75. Then, we look up the corresponding probability from the standard normal distribution table and subtract it from 1 since we want the probability of being greater. The value from the table is 0.9599, so P(X > 13.5) = 1 - 0.9599 = 0.0401.
For part (c), we need to find P(X < 9.4). Using the same standardization process, we calculate z = (9.4 - 10) / 2 = -0.3. Looking up the corresponding probability from the standard normal distribution table, we find it to be 0.3821. Therefore, P(X < 9.4) = 0.3821.