Final answer:
The mean of the sample mean is 25, identical to the population mean, and the standard error of the sample mean for samples of size 64, drawn from a population with a standard deviation of 28, is calculated as 3.5.
Step-by-step explanation:
The question concerns the calculation of the standard error of the sample mean and its distribution when samples are taken from a population with known mean and standard deviation. To find the standard error of the sample mean, we use the formula:
Standard Error = \( \frac{{\sigma}}{{\sqrt{n}}} \)
where \( \sigma \) is the population standard deviation and \( n \) is the sample size.
In this scenario, the population standard deviation (\( \sigma \)) is 28, and the sample size (\( n \)) is 64. The calculation is as follows:
Standard Error = \( \frac{{28}}{{\sqrt{64}}} \) = \( \frac{{28}}{{8}} \) = 3.5.
The mean of the sample population remains the same as the population mean, which is 25. Therefore, the distribution of the sample mean would be normally distributed with a mean of 25 and a standard error of 3.5 if the population is normal or if the sample size is large enough (by the Central Limit Theorem).
The correct answer for the mean and standard error of the sample mean in this scenario is choice b) 25 and 3.5.