13.1 The central 95% of any Normal distribution lies within two standard deviations of the mean. Two standard deviations is 5 inches here, so the middle 95% of the young men’s heights is between 65 inches (that’s 70 − 5) and 75 inches (that’s 70 + 5).

13.2 The standard score of a height of 72 inches is

13.3 To fall in the top 25% of all scores requires a score at or above the 75th percentile. Look in the body of Table B for the percentile closest to 75. We see that a standard score of 0.7 is the 75.80 percentile, which is the percentile in the table closest to 75. So, we conclude that a standard score of 0.7 is approximately the 75th percentile of any Normal distribution.

To go from the standard score back to the scale of SAT scores, “undo” the standard score calculation as follows:

observation = mean + standard score × standard

deviation

= 500 + (0.7) (100) = 570

A score of 570 or higher will be in the top 25%.

For scores at or below 475:

Looking up a standard score of 20.3 in Table B, we find 38.21% of scores will be at or below 475.

For scores at or above 580:

Looking up a standard score of 0.8 in Table B, we find 78.81% of scores will be below 580. Taking the complement, 100% − 78.81%, we find that 21.19% of scores will be at or above 580.