Download add the variances - Madison Public Schools

AP Statistics
Notes 6.3
Let T = X + Y
µT = µX + µY (you can add the means of different random variables)
𝜎𝑇2 = 𝜎𝑋2 + 𝜎𝑌2
(you can add the variances of different random variables)*
Let D = X – Y
µD = µX - µY (you can subtract the means of different random variables)
𝜎𝐷2 = 𝜎𝑋2 + 𝜎𝑌2
(you MUST still add the variances of different random
variables)*
* only true for independent random variables!
1. Using information from CCM, the number of credits X that a randomly selected full-time
student is taking in the fall semester has µx = 14.65 and σx = 2.06. The number of credits Y that
a randomly selected full-time student is taking in the spring semester has µy = 13.96 and σy =
2.00. What is the mean number of credits that a full-time student takes at CCM during the fall
and spring semester? What is the standard deviation of the sum?
2. Using the information from the previous problem, what is the mean difference of the number of
credits that a full-time student takes at CCM for the fall and spring semester? What is the
standard deviation of the difference? This time, assume that the number of credits taken in each
semester is independent.
3. To save time and money, many single people have decided to try speed dating. At a speed
dating event, women sit in a circle and men spend about 5 minutes getting to know a woman
before moving on to the next one. Suppose that the height M of male speed daters follows a
Normal distribution with a mean of 70 inches and a standard deviation of 3.5 inches and the
height F of female speed daters follows a Normal distribution with a mean of 65 inches and a
standard deviation of 3 inches. What is the probability that a randomly selected male speed
dater is taller than the randomly selected female speed dater with whom he is paired?
1. State: What is the probability that a randomly selected male speed dater is taller than the
randomly selected female speed dater with whom he is paired?
Plan: We’ll define the random variable D = M – F to represent the difference between the
male’s height and the female’s height. Our goal is to find P(M > F) or P(D > 0).
Do: Since D is the difference of two independent Normal random variables, D follows a
Normal distribution with mean D  M  F = 70 – 65 = 5 inches and variance  D2   M2   F2
= 3.52 + 32 = 21.25. Thus, the standard deviation is  D  21.25 = 4.61 inches.
Thus, P(D > 0) = normalcdf(0, 99999, 5, 4.61) = 0.8610. Note: to get full credit on the AP
exam when using the calculator command normalcdf, students must clearly identify the shape
(Normal), center (mean = 5) and spread (standard deviation = 4.61) somewhere in their work.
Conclude: There is about an 86% chance that a randomly selected male speed dater will be
taller than the female he is randomly paired with. Or, in about 86% of speed dating couples,
the male will be taller than the female.