Area Under the Sampling Distribution of the Mean (1 of 4)

Assume a test with a mean of 500 and a standard deviation of 100. Which is more likely: (1) that the mean of a sample of 5 people is greater than 580 or (2) that the mean of a sample of 10 people is greater than 580? Using your intuition, you may have been able to figure out that a mean over 580 is more likely to occur with the smaller sample.

One way to approach problems of this kind is to think of the extremes. What is the probability that the mean of a sample of 1,000 people would be greater than 580. The probability is practically zero since the mean of a sample that large will almost certainly be very close to the population mean. The chance that it is more than 80 points away is practically nil. On the other hand, with a small sample, the sample mean could very well be as many as 80 points from the population mean. Therefore, the larger the sample size, the less likely it is that a sample mean will deviate greatly from the population mean. It follows that it is more likely that the sample of 5 people will have a mean greater than 580 then will the sample of 10 people.