Final answer:
To calculate the probabilities associated with the new sample's mean relative to the population mean, we need to use the Z-score formula and refer to the standard normal distribution. The specific probabilities require knowledge of the population standard deviation or standard error.
Step-by-step explanation:
The student's question pertains to the probability of sample means from a normally distributed population. The population has a given mean (population means), and we are interested in calculating different probabilities associated with the sample means of size n = 25.
a. To find the probability that the new sample's mean is at least 6, we would use the Z-score formula. However, to compute this, the value of the population standard deviation or the standard error must be known. Assuming we have this information, we would calculate the Z-score for a mean of 6 and use the standard normal distribution to find the corresponding probability.
b. Similarly, the probability that the sample's mean falls between 3 and 7 would involve finding the Z-scores for both values and using the standard normal distribution to find the probability that lies between these Z-scores.
c. Finding the probability of the sample mean falling within 1 and 2 standard errors below the population mean also requires the standard error. We would find the Z-scores corresponding to one and two standard errors below the mean and again refer to the standard normal distribution for the probabilities.