Final answer:
The question inquires about the sampling distribution of the sample mean from a population with known mean and standard deviation, and asks for the probability that the sample mean falls within a specific range. The sample mean is equal to the population mean, while the sample's standard error must be calculated. The probability is found using the properties of the normal distribution and the Central Limit Theorem.
Step-by-step explanation:
The student is asking a question within the subject of statistics, specifically about the sampling distribution of the sample mean.
The distribution of the variable x is given with a mean (μ) of 24 and a standard deviation (σ) of 19.
When a sample of size n = 48 is drawn, the sample mean μx will remain the same as the population mean, so μx = 24.
However, the standard error of the mean σx will be σ/√n = 19/√48 which needs to be calculated to give us the spreading of the sampling distribution.
The probability P(24 ≤ x ≤ 26) can be found using the standard normal distribution since the sample size is large.
Because the sample size is greater than 30, we can apply the Central Limit Theorem which allows us to assume that the sampling distribution of the mean will be approximately normal, even if the population distribution is not.
We can use this property to find the probability that the sample mean falls within a certain range by standardizing and finding the z-scores corresponding to x = 24 and x = 26, and then looking up these z-scores in a standard normal distribution table or using a calculator.