Final answer:
The sample mean will be within 8 centimeters of the population mean 68% of the time, calculated using the standard error formula and the empirical rule for a normal distribution. The correct answer is option B.
Step-by-step explanation:
To calculate how many centimeters the sample mean will likely be within the population mean 68% of the time, we can use the concept of standard error and apply the empirical rule, which is relevant to normally distributed data. The standard error (SE) of the mean is given by the formula SE = σ / √n, where σ is the population standard deviation and n is the sample size.
Given a population standard deviation (σ) of 128 centimeters and a sample size (n) of 256, the standard error is calculated as follows:
SE = 128 / √256
SE = 128 / 16
SE = 8 centimeters
According to the empirical rule, for a normal distribution, 68% of the values are within one standard deviation from the mean. Therefore, the sample mean will be within 8 centimeters of the population mean 68% of the time. The correct answer is B. 8.