Question 767:

Asked on June 24, 2009

Tags: Z-Score

1

Answer:

No answer provided yet.So, the concept here is to change the z-score formula just a bit to take into account the sample size. The z-score for a point (instead of an average) is (x-mean)/standard deviation. To take into account the sample size we divide by the standard error of the mean (SEM) which is the sample standard deviation divided by the square root of the sample size.

In this example, our SEM is 15/SQRT(25) = 3.  We now substitute it into the z-score formula for the mean of 100 to find the lower of the two values first: (95-100)/3 = -1.6667. To find the probability of this value we use a normal table or the z-score to percentile calculator and enter -1.6667 for 1 sided area. We get .0477.  We're half way done. Now we need to find the probability of the 105 score. Using the same approach we get  (105-100)/3 = 1.6667. We get the probability of  .95221 will be less than 105.

Finally, to find the area in between these two values we subtract the lower value from the large value = .92221-.04779 = .904419. So the probability the mean score of 25 people is between 95 and 105 is .904419 or about 90%.