Final answer:
For a population with a normal distribution, the distribution of the sample mean of a sufficiently large sample size is true to have a mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size. Option 1 is the correct answer.
Step-by-step explanation:
The question pertains to the distribution of sample means within the context of the Central Limit Theorem. For a population characterized by a normal distribution with mean μ (µ) and standard deviation σ (σ), the distribution of the sample mean (¯X) from a sufficiently large sample size (n) is also normally distributed. The mean of this distribution is equal to the population mean (µ), and the standard deviation of the sample means, often referred to as the standard error, is σ divided by the square root of n (σ/√n).
This concept is crucial in inferential statistics, as it allows us to estimate the population mean from sample means. As per the Central Limit Theorem, regardless of the population's distribution, the distribution of sample means will approximate a normal distribution when the sample is large enough. This theorem applies to sample means and sample sums used in statistics. The empirical rule indicates that approximately 95 percent of samples will have the sample mean (¯X) within two standard deviations of the population mean (µ).
Therefore, the statement for the sample mean ¯X being normally distributed with a mean equal to the population mean (µ) and a standard deviation equal to the population standard deviation σ divided by the square root of the sample size n (σ/√n) is true.