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%
