Given that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2, find the probability that X is between 45 and 54.
To find the probability that X is between 45 and 54, we can first standardize these values using the z-score formula:
Z = (X - ยต) / ฯ
For X = 45:
Z1 = (45 - 50) / 2
Z1 = -2.5
For X = 54:
Z2 = (54 - 50) / 2
Z2 = 2
Next, we can use a standard normal distribution table (or a calculator) to find the probabilities associated with these z-scores:
P(Z < -2.5) = 0.0062
P(Z < 2) = 0.9772
To find the probability that X is between 45 and 54, we need to subtract the probability of Z < -2.5 from the probability of Z < 2:
P(45 < X < 54) = P(Z < 2) - P(Z < -2.5)
P(45 < X < 54) = 0.9772 - 0.0062
P(45 < X < 54) = 0.971
Therefore, the probability that X is between 45 and 54 is 0.971, or 97.1%.