How large a smaple from a N(80,20) population is needed so that the sampling distribution of the mean has a standard deviation of 1?

To determine the sample size needed for a specific standard deviation of the sampling distribution of the mean, we can use the formula:

n = (Z * σ) / E

where:
n = sample size
Z = the z-score corresponding to the desired level of confidence (e.g., for a 95% confidence level, the z-score is approximately 1.96)
σ = standard deviation of the population
E = margin of error (which is the desired standard deviation of the sampling distribution of the mean)

In this case, you want the standard deviation of the sampling distribution of the mean to be 1. So, E = 1.

Given that the population has a normal distribution N(80,20), with a standard deviation of 20. We can plug in the values into the formula to calculate the sample size.

n = (Z * σ) / E
n = (1.96 * 20) / 1
n ≈ 39.2

Therefore, a sample size of approximately 39 or more would be needed to ensure that the sampling distribution of the mean has a standard deviation of 1.