standard diciation

Question

Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.542.54 and a standard deviation of 0.420.42. Using the empirical rule, what percentage of the students have grade point averages that are less than 2.122.12? Please do not round your answer.

Answer 1

Hi Faith,

The empirical rule is also known as the 68-95-99.7 rule. This means that if data is normally distributed i.e. has a bell-shaped curve, 68% of that data falls within one standard deviation of the mean, 95% falls within two standard deviations of the mean, and 99.7% falls within three standard deviations of the mean. So:

Mean=2.54254

SD=0.42042

You want students with GPAs below 2.12212, so:

2.4254-(0.42042)=2.12212=One Standard Deviation Below Mean

You can now eliminate 68% of your students from the population, since they fall within one standard deviation either way. That leaves you with 32% of the students to account for. But, remember that half of these students will fall above the mean, and you don't want these, so your answer is 16%.

Alternatively, if you use z-scores and compute z=x-mu/sigma where:

x=2.12212

mu=mean=2.54254

sigma=sd=0.42042

z=-1; from table at z-table.com:

P=0.1587, 15.87%

Not sure which answer your instructor prefers, but both work.