Final answer:
The probabilities for both a single randomly selected value and a sample mean for a normally distributed population can be calculated using the individual standard normal distribution and the Central Limit Theorem respectively.
Step-by-step explanation:
The student's question involves calculating probabilities related to normal distributions and the Central Limit Theorem. With a known population mean and standard deviation, we aim to find the probability for a single value and the probability of sample means with given characteristics.
For part A, we can use the normal distribution with the population mean (μ) and standard deviation (σ) to find the probability that a single value is less than 175. For part B, to find the probability that the sample mean of size 13 is less than 175, we apply the Central Limit Theorem, which states that the sampling distribution of the sample mean is also normally distributed with mean μ and standard deviation σ/√n, where n is the sample size.
Using similar logic, part C's probability can be found by subtracting part A's probability from 1, since it's the complementary event. Part D's probability is the complement of part B's, found by applying the Central Limit Theorem and subtracting the resulting probability from 1.