The Bell Curve aka Normal Distribution aka Gaussian is important stuff that everybody functioning in modern society should understand at least up to the 68-95-99.7 rule.

A normal distribution is characterized by a mean μ and a standard deviation σ. The distribution has the characteristic bell shape, with the center of the bell at x=μ.

The value of σ says how wide the bell is. That's where the 68-95-99.7 rule comes in. The first value, 68, means that 68% of the area of the bell curve is contained in plus or minus one standard deviation. That means there's a 68% chance a random x drawn from this distribution is between μ-σ and μ+σ.

The 95 means that 95% of the area of the bell within two standard deviations of the mean. A random x has a 95% chance of being between μ-2σ and μ+2σ.

It's the same story for 99.7% except that encompasses everything within three standard deviations of the mean.

Now let's answer the question. We have μ=10, σ=1.5.

a. We want a 95% probability, which we learned is within two standard deviations, two sigma of the mean. So the lower bound is 10 - 2(1.5) = 7 and the upper bound is 10 + 2(1.5) = 13.

Answer: **7 to 13**

b. We're asked for 68%; we know that's one sigma, from 10-1.5 to 10+1.5.

Answer: **8.5 to 11.5**

4.

μ=72, σ=2

a.

We want P(x < 68)

In general to do these sorts of problems we convert our x test on the particular normal distribution given to a z test on a standard normal distribution with mean zero and standard deviation one.

z = (x - μ)/σ = (68 - 72)/2 = -2

We're interested in P(z < -2), i.e. the probability we landed more than two standard deviations below the mean. The 68-95-99.7 rules says 95% is between plus or minus two sigma, which leaves 5%, split equally between being less than minus two sigma and more than plus two sigma.

Answer: **2.5%**

b.

I forgot to answer part b. Here we go.

Between 70 and 72 inches is between -1 standard deviation below the mean and the mean. So we want the area of the bell curve between z=-1 and z=0. That's half of the 68% that we get when we go from z=-1 to z=1.

Answer: **34%**