The Empirical Rule Theorem states that:

– 68% of the observations lie within one standard deviation of the mean.

– 95% of the observations lie within two standard deviations of the mean.

– 99.7% of the observations lie within three standard deviations of the mean.

And just to clarify:

Mean or (µ) is just another word for “average”.

Heres an example I made up:

A sample of the cars traveling along the Tri-City Highway approximates a symetrical, bell-shaped distribution.

Suppose the mean (or average) speed is 60 mph

with a standard deviation (σ) of 5 mph.

1) What is the speed of 68% of the cars?

Answer: 55 to 65 mph (60 ± 5 mph)

2) What is the speed of 95% of the cars?

Answer: 50 to 70 mph (60 ± 10 mph)

3) What is the speed of 99.7% of the cars?

Answer: 45 to 75 mph (60 ± 15 mph)

——————-

Back to your question:

(a) Between what heights do the middle 95% of young women fall?

Answer:

= 64 inches ± 2 standard deviations (σ)

= 64 in. ± 2σ

= 64 in. ± 2 * (2.7 in.)

= 64 in. ± 5.4 in.

= 58.6 in. to 69.4 in.

(b) What percentage of young women are taller than 61.3 inches?

We know that 50% of the women are above the mean, right?

And 61.3 inches is one standard deviation BELOW the mean.

64 – 61.3 = 2.7 (one standard deviation)

So, the answer is:

50% + (1/2 * 68%) = 84%

This website gives a short, but very good explanation:

http://www.oswego.edu/~srp/stats/6895997.htm

Good luck in your studies,

~ Mitch ~

P.S. – It’s best to ‘visualize’ this problem with this picture…

http://www.oswego.edu/~srp/stats/images/normal_34.gif