The heights of German men are normally distributed with a mean of 65.5 inches and a standard deviation of 3.6 inches.

Question

According to the empirical rule (68-95-99.7 Rule), what percentage of German men are:

 Between 58.3 and 72.7 inches tall?

Answer: ____  %

(b) Over 69.1 inches tall?

Answer: ____ %

(c) Under 67.912 inches tall?

Answer: ____ %

Just confused as to how you solve this problem.

Answer 1

Okay, this is backwards, but if you prefer start at the bottom with the a) part first

c) 79.6% using a z-table with z value of 2.41/3.6=slightly more than 2/3 standard deviation from the mean. You can't use the 68-95-99 rule directly, but you can estimate or approximate the percentage. From part b) below you know slightly more than 17% are above 67.912, leaving 100%-17%=83% below 67.912. Round it off to 80% and you should get very close to that with the z-table or you may want to recheck it. z-tables will get you between 80% and 79% but much closer to 80%. Interpolate and it's 79.6% Or use a scientific calculator with statistics programs. Or google a z-table calculator.

b) 17%, using the 68-95-99 rule. 69.1-65.5=3.6 or exactly one standard deviation from the mean. 68% are within plus or minus one standard deviation from the mean. That means 100%-68%=34% are more than or less than one standard deviation from the mean, with half that 17% above one standard deviation from the mean. The other 17% is less than one standard deviation from the mean

a) 95% of German men are within 2 standard deviations from the mean. 72.7-65.5=7.2=2 times 3.6

same with the lower bound, 65.5-58.3=7.2 or 2 standard deviations below the mean.

The 99.7 part never got involved in this problem. Sometimes they just call it 99, shortening it from 99.7. Either way nearly 100% of all the data falls within 3 standard deviations of the mean.