Expert: Bobby Soltani - 8/5/2004

Question
On a standardized test with a normal distribution and a standard deviation of 2, a score of 35 will occur less than 5% of the time. Which of the following could be the mean for the test?
1. 32
2. 34
3. 38
4. 40
The answer is 40, but can't get it. Please help.  

Answer
Ab,

Please read the following paragraph carefully before I explain this problem.

One useful characteristic of a normal curve is that certain percentages of scores fall at certain distances (measured in standard deviation units) from its mean. A special statistical table makes it a simple matter to determine the percentage of scores that fall above or below a particular score or between two scores on the curve. For example, about 68 percent of scores fall between plus and minus one standard deviation from the mean; about 95 percent fall between plus and minus two standard deviations from the mean; and about 99 percent fall between plus and minus three standard deviations from the mean.
(pasted from http://www.mhhe.com/socscience/intro/cafe/common/stat/dstats10.mhtml)

After reading that statement, we can interpret what the question is saying.  It is saying that a score of 35 is at least 2 standard devations away from the mean since it occurs less than 5 percent of the time.  Since one standard deviation is equal to 2, two standard deviations is equal to 4.  That means that the mean must be at least 4 away from 35.  In otherwords the mean cannot be in the range 31-39 (which is 35 plus or minus 4).  The only answer outside this range is 40.

I hope this helps your understanding.

Bobby