Okay. I will list all the relatively prime numbers up to 331. 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101, 103,107,109,113,127,131,137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331. Okay, so look at this list and see which match up. For A. 102 and 312. Neither of these numbers are relatively prime. For B. 10 and 45. Neither of these are relatively prime. For C. 3 and 51. 3 is a relatively prime number, 51 is not. For D. 35 and 72. Neither of these are relatively prime numbers. But the answer would be D. because to get a relatively prime pair of numbers you have to have both of them not be divisible by the same number. 102 and 312 can be divided 2, so that's not the answer. 10 and 45 can be divided by 5, so that is incorrect. 3 and 51 can be divided by 3, so that is also incorrect. 35 and 72 cannot be divided by the same numbers. So, the answer is D. 35 and 72.
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.
