A researcher selects a sample of 25 participants from a population with a mean of 10 and a standard deviation of 5.What is the probability of selecting a sample mean that is at least two standard deviations larger than the population mean?
A) .0228
B) .4772
C) .9772
D) equal to the probability of selecting a sample mean that is at least two standard deviations below the mean

Question

Quizplus · Accepted Answer

The probability of selecting a sample mean that is at least two standard deviations larger than the population mean is found using the standard normal distribution (Z-distribution). The Z-score for two standard deviations above the mean is 2. The Central Limit Theorem allows us to treat the distribution of sample means as normally distributed if the sample size is sufficiently large (n ≥ 30 is a common rule of thumb, but n = 25 is often considered large enough for the approximation to be useful). The standard error of the mean (SEM) is the standard deviation of the population divided by the square root of the sample size. In this case, the SEM = 5 / sqrt(25) = 1. The Z-score for a sample mean that is two standard deviations (10 units in this context, since the population standard deviation is 5 and we're considering 2 of these) above the population mean of 10 is (20 - 10) / 1 = 10. However, since we're looking for the probability of being at least two standard deviations above, we're interested in the tail beyond Z = 2. Looking up the Z-score of 2 in standard normal distribution tables or using a calculator gives us the area to the left of Z = 2, which we subtract from 1 to find the tail probability. This gives us approximately 0.0228, indicating a small probability of selecting such a sample mean by chance.

A Researcher Selects a Sample of 25 Participants from a Population