Question

If we had a population that was normally distributed and had a mean of 50 and a standard deviation of 5….

a. If we sampled a lot of items, say 100 items, from this population, what should we expect the largest and the smallest values to be in that sample? (Use the empirical rule to come up with your answer.)

b. Now assume we repeated the experiment in part “a” 100 times….In other words, we took 100 subgroups of 100 items. We would expect the mean of the means to be equal to the mean of the individuals, but due to random variation it is unlikely that the two would actually be the same. So, what should be the largest and smallest means of the 100 subgroups?

Answer 1

a. As we know the empirical rule of 68 - 95 - 99.7 that is the percent value that are one, two or three standard deviation away values from the mean.

so here that means that the values at the three standard deviation away could be expect to be the largest and smallest values in the sample.

That would be

smallest value = mean - 3 * standard deviation = 50 - 3 * 5 = 35

Largest value = mean + 3 * standard deviation = 50 + 3 * 5 = 65

b. Now we take 100 samples of size 100.

so here standard error of sample mean = standard deviation/sqrt(sample size) = 5/sqrt(100) = 0.5

so here expected largest mean and smalles mean would be three standard error away from the expected population mean that is 50.

so here expected smallest sample mean value = mean - 3 * standard error = 50 - 3 * 0.5 = 48.5

largest sample mean value = mean + 3 * standard erorr = 50 + 3 * 0.5 = 51.5