Notes 7.4b
Shifting and Combining Random Variables
Name___________________________
1. The test scores for 5 randomly selected students are 80, 92, 95, 78, 86. Find the mean, standard deviation, variance
and range of the scores.
Mean=__________ s=__________ var=___________ range(high-low)=_________
b. Add 5 points to to each score. Then find the mean, standard deviation, variance and range of the scores.
Mean=__________ s=__________ var=___________ range=_________
c. Increase each data point by 15%. Then find the mean, standard deviation, variance and range of the scores.
Mean=__________ s=__________ var=___________ range=_________
What do you notice?
Shifting data……
Adding to each data value, shifts the measures of location, not the measures of spread.
Multiplying each data value, shifts the measures of location and the measures of spread.
General Form: Start with a random variable X.
aX + b
1 Random Variable
.
aX – b
.
Mean:
Sd:
Example: Ch. 6 Test Scores Per 1(X): Mean = 82 s = 8.1
a) Find the mean and sd for 3X + 6
2 INDEPENDENT
X+Y
.
X – Y.
.
Random Variables
Mean:
Sd:
Suppose we have Per 2(Y): Mean = 75 s = 15.4
a) Find the mean and sd for X – Y
b) Find the mean and sd for 2X + 4Y
1. Scores on the Mathematics part of the SAT college entrance exam in a recent year had mean 611 and standard
deviation 110. Scores on the Verbal part of the SAT had mean 501 and standard deviation 102.
(a) What is the mean of the total SAT score (Math plus Verbal)?
(b) If you can calculate the standard deviation of the total SAT score, do it. If not, explain clearly why you can’t.
You Try:
2. Given independent random variables, with means and standard deviations as shown,
find the mean and standard deviation of each of these variables:
a) X – 40
b) 1.2Y
c) 3X + 25
d) X + Y
e) 2X – 5Y
f) X – 1.5Y
Mean
SD
X
70
11
Y
14
5
3. Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale
with possible values of 1 to 5. Assume that X is a random variable representing the pain score for a randomly
selected patient. The following table gives part of the probability distribution for X.
X 1
P(X) 0.2
2
0.4
3
0.1
4
0.1
5
?
a) Find the mean µ for this distribution.
b) Find the variance and standard deviation for this distribution.
c) Suppose the pain scores for two randomly selected patients are recorded. Let Y be the random variable
representing the difference of the two scores. Find the mean and standard deviation of Y. Assume the selected
patients are independent of one another.
4. On a small commuter plane baggage weight is a concern because it effects how quickly the plane can ascend.
Suppose x = weight of a randomly checked bag has a mean of 36 and a standard deviation of 11. Consider a flight
on which 6 passengers, all traveling alone, are flying. If xi is the baggage weight for passenger i, (for i ranging from
1 to 6), the total weight of checked baggage is then:
Y = x1 + x2 + . . . +x6
a) Find the mean, the variance and the standard deviation of Y.