Final answer:
To solve the problem, calculate the z-score for both individual and sample heights. There is a 56.46% chance that a randomly selected woman is under 64 inches tall, and a 78.81% chance that the average height of 25 women is less than 64 inches.
Step-by-step explanation:
The question concerns normal distribution and probability in the context of statistics, a field of mathematics. The steps to answer each part of the question are as follows:
a. To find the probability that a randomly selected woman has a height of less than 64 inches, calculate the z-score and then use the standard normal distribution table or a calculator with statistical functions. The z-score is calculated as follows:
Z = (X - μ) / σ
Where X is 64 inches (the height we're interested in), μ is the mean height (63.6 inches), and σ is the standard deviation (2.5 inches). Substituting the values, we get:
Z = (64 - 63.6) / 2.5
Z = 0.4 / 2.5
Z = 0.16
After finding the z-score, refer to the z-table to find the probability. For a z-score of 0.16, it's approximately 0.5646. Since the z-table gives us the probability that the variable falls to the left of a given z-score, it means there is approximately 56.46% chance that a randomly selected woman's height is less than 64 inches.
b. For the second part, where 25 women are randomly selected, use the standard error (SE), which is the standard deviation divided by the square root of the sample size (n), to account for the sampling distribution of the sample mean:
SE = σ / √n = 2.5 / √25 = 2.5 / 5 = 0.5 inches
Then calculate the z-score using the sample's mean rather than an individual's height:
Z = (μ_{sample} - μ) / SE
Z = (64 - 63.6) / 0.5
Z = 0.8
Looking up a z-score of 0.8 gives us a probability of approximately 0.7881. Therefore, there is about a 78.81% chance that the average height for a sample of 25 women is less than 64 inches.