Final answer:
To calculate the probability of a height being less than a certain value in a normal distribution, we use z-scores. We find the z-scores by subtracting the mean from the value and dividing by the standard deviation, then look up these scores in a z-table or use a calculator.
Step-by-step explanation:
The question asks about calculating probabilities from a normal distribution, which is a common task in statistics, a branch of mathematics. Specifically, it requires us to apply our understanding of the properties of the normal distribution to find probabilities based on given means and standard deviations.
To find the probability that a study participant has a height that is less than 66 inches, we use the standard normal distribution and the concept of z-scores. A z-score is calculated by subtracting the mean from the given value and dividing the result by the standard deviation:
z = (X - μ) / σ
Substitute the given values:
z = (66 inches - 69.7 inches) / 3.0 inches = -3.7/3 = -1.23
We then use the z-score to find the probability from standard normal distribution tables or a calculator equipped with statistical functions.
For part b, the probability that a participant's height is between 66 and 72 inches, we find the z-scores for both values, look up the corresponding probabilities, and subtract the smaller probability from the larger one.