Final answer:
To find the probability that a randomly chosen x from a population with a normal distribution is between two values, calculate the area under the normal distribution curve between those values using z-scores.
Step-by-step explanation:
To find the probability that a randomly chosen x from the population is between 11.9 and 13.5, we need to calculate the area under the normal distribution curve between these two values. Since the population has a normal distribution with a mean of 12.7 and a standard deviation of 1.4, we can use the standard normal distribution to calculate the probability.
First, we need to convert the values of 11.9 and 13.5 to their z-scores. The z-score formula is z = (x - µ) / σ, where x is the value, µ is the mean, and σ is the standard deviation.
Calculating the z-scores:
z1 = (11.9 - 12.7) / 1.4 = -0.571
z2 = (13.5 - 12.7) / 1.4 = 0.571
Next, we can use a standard normal distribution table or a calculator to find the area under the curve between these z-scores. The area between -0.571 and 0.571 is approximately 0.427. Therefore, the probability that a randomly chosen x from the population is between 11.9 and 13.5 is 0.427, or 42.7%.