mu = mean = 25
sigma = standard deviation = 5.5
Compute the z score for x = 19.5
z = (x-mu)/sigma
z = (x-25)/5.5
z = (19.5-25)/5.5
z = -1
Compute the z score for x = 30.5
z = (x-mu)/sigma
z = (x-25)/5.5
z = (30.5-25)/5.5
z = 1
Therefore, P(19.5 < X < 30.5) is the same as P(-1 < Z < 1) which asks "what is the area under the standard normal curve from z = -1 to z = 1?"
By the Empirical Rule, roughly 68% of the normal distribution is between z = -1 and z = 1. In other words, 68% of the normally distributed population is within one standard deviation. The z-scores help keep track how far you are in terms of standard deviations.
Final Answer: Choice B) 68%
