## Question 572:

1

1. We need to assume that the GPAs come from a normally distributed population. With a normal distribution we can simply use the properties of the normal curve.

2. The empirical rule states that approximately 68% of the values fall within 1 standard deviation around the mean, and 95% of the values fall within ~2 standard deviations of the mean. This refers to the values above and below the mean. In other words, 16% would fall above 1 standard deviation and 16% would fall below. Likewise, ~2.5% would fall above 2 standard deviations and 2.5% would fall below 1 standard deviation below the mean.  So, the z-score limits should be set at approximately 1 and 2. We can double check our answers by looking the values up in a normal table or using the percentile to z-score calculator and entering .84 and .975 and selecting 1-sided area. We get the values of .9944 and 1.96--which are pretty close to 1 and 2. We used .84 and .975 instead of .16 and .025 since we want the top percentile scores, not the bottom percentile scores.

3. The GPAs corresponding to 1 and 2 standard deviations above the mean can be found by simply adding 1 and 2 standard deviations to the mean. The standard deviation is given as 0.5. The mean is 2.7 so 1 standard deviation above is 2.7+0.5= 3.2 and 2.7+.05+.05 = 3.7. So the GPAs would need to be 3.2 and 3.7 for magna and summa cum laude respectively