Rules for Means and Variances
 Rules for Means
o If X is a random variable and a and b are constants, then
 ma +X = a + mX
 mbX = bmX
 ma +bX = a + bmX
o If X and Y are random variables, then mX +Y = mX + mY
 Rules for Variances
o If X is a random variable and a and b are constants, then
 s a2+X = s X2
2
 sbX
= b2sX2
 sa2+bX = b2sX2
o If X and Y are independent random variables, then
 sX2 +Y = sX2 + sY2
 sX2 -Y = sX2 + sY2
Linear Transformations:
1. In a large introductory statistics class, the distribution of X = raw scores on a
test was approximately normally distributed with a mean of 17.2 and a
standard deviation of 3.8. The professor decides to scale the scores by
multiplying the raw scores by 4 and adding 10.
a) Define the variable Y to be the scaled score of a randomly selected
student from this class. Find the mean and standard deviation of Y.
b) What is the probability that a randomly selected student has a scaled test
score of at least 90?
Rules for Means and Variances of a Linear Transformation of a Random
Variable
1. The mean of the Ch. 5 Test was 133 points with a variance of 36. Suppose that
the teacher gives each student an additional 5 points with the hope that it will
make students like her more. Find the new mean and variance of the test scores.
Formulas used: _______________________, _______________________
2. Suppose that instead of giving each student an additional 5 points, she decides to
multiply each student’s score by 1.1. Find the new mean and variance of the test
scores.
Formulas used: _______________________, _______________________
3. A candy manufacturer makes nut cluster bars and fudge. Of course not every nut
cluster bar has the same weight and the same is true of different pieces of fudge.
The manufacturing process for each candy is designed so that the mean weight
of nut cluster bars is 4 ounces with a variance of .04 ounces and the mean weight
of pieces of fudge is 6 ounces with a variance of .25 ounces. The company sells a
gift box containing three nut cluster bars and four pieces of fudge. What is the
mean of the gift box’s weight?
Formula used: _______________________
(over)
4. At a particular school, on the SAT Math test the school mean was 510 with a
standard deviation of 102, while on the SAT Verbal test the school mean was 490
with a standard deviation of 110. A student’s composite score is the sum of
his/her scores on the math and verbal sections of the test. What are the mean
and standard deviation of composite scores at this school? (Note: you’ll have to
convert the standard deviations to variances and then back again.)
Formulas used: _______________________, _______________________
5.