Chapter 7: RV's & Probability Distributions
Review Packet
Name _________________________
The following questions are in a True / False format. The answers to these questions will frequently depend on
remembering facts, understanding of the concepts, and knowing the statistical vocabulary. Before answering
these questions, be sure to read them carefully!
T
F
1. A random variable is discrete if the value of the random variable depends
upon the outcome of a chance experiment.
2. A random variable is continuous if the set of possible values includes an
entire interval on the number line
3. The distribution of all values of a random variable is called a normal
distribution.
4. For a continuous random variable x, the height of the density curve over an
interval a to b represents the probability that x is between a and b.
T
F
T
F
T
F
T
F
5. For every random variable, P  a  x  b   P  a  x  b  .
T
F
6. For a discrete random variable x,  x 

x  p  x
x values
T F
7. For a discrete random variable,  x 
  x     p  x
x
x values
T
F
8. If x is a random variable, and random variable y is defined as follows,
y = a + bx , then my = b s x .
T
F
9. For random variables, x and y, if y  a  bx , then  y  a  b x
T
F
10. If random variables x1 and x2 are independent, then  x2  x   x2   x2
1
2
1
2
T
F
11. The standard normal distribution has a mean of 1 and standard deviation of
0.
T
F
12. A normal probability plot suggests that a normal probability model is
plausible if there is no obvious pattern in the scatter of points.
T
F
13. The probability of k successes among n independent trials, each with equal
n- k
probability of success p , is: n Ck p k (1 - p ) .
T
F
14. The probability of the first success at trial x of a sequence of independent
trials, each with equal probability of success p , is: p x- 1 (1- p ) .
Chapter 7, Review
Page 1 of 10
Chapter 7: RV's & Probability Distributions
Section 7.1-7.3
1. What is a random variable?
2. Using the notation C = continuous and D = discrete, indicate whether each of the random variables are
discrete or continuous.
a) The number of defective lights in your school's main hallway
b) The barometric pressure at midnight
c) The number of staples left in a stapler
d) The number of sentences in a short story
e) The average oven temperature during the cooking of a turkey
f) The number of lightning strikes during a thunderstorm
Chapter 7, Review
Page 2 of 10
3. At the College of Warm & Fuzzy, good grades in math are very easy to come by. The grade distribution is
given in the table below:
Grade
Proportion
A
0.30
B
0.35
C
0.25
D
0.09
F
0.01
Suppose three students are to be selected at random. As each is selected their math grades are written down
and they are replaced back into the population of students. Three possible outcomes of this experiment are
listed below. Calculate the probabilities of these sequences appearing.
a) BAC
b) CFF
c) ABA
4. The famous physicist, Ernest Rutherford, was a pioneer in the study of radioactivity using electricity. In one
experiment he observed the number of particles reaching a counter during time 1700 intervals of 7.5
seconds each. The number of intervals that had 0 – 4 particles reaching the counter is given in the table
below.
k particles
Number of time
intervals with k
particles
0
1
2
3
4
57
203
383
525
532
Chapter 7, Review
Page 3 of 10
Let the random variable x = number of particles counted in a 7.5 second time period.
a) Fill in the table below with the estimated probability distribution of x, and sketch a probability histogram for
x.
Probability distribution
x
Probability histogram
P(x)
0
1
2
3
4
b) Using the estimated probabilities in part (a), estimate the following:
i) P(x = 1), the probability that 1 particle was counted in 7.5 seconds.
ii) P(x < 3) the probability that fewer than 3 particles were counted.
iii) P(x ³ 3) the probability that at least 3 particles were counted.
Chapter 7, Review
Page 4 of 10
5.
The density curve for a continuous random variable is shown below. Use this curve
to find the following probabilities:
0.5
1.0
2.0
You may use the following area formulas in your calculations:
Area of a rectangle: A  lw
1
Area of a trapezoid: A  h  b1  b2 
2
1
Area of a right triangle: A  ab
2
a) P  x  1
b) P  2  x  3
c) P  x is at least 3
Chapter 7, Review
Page 5 of 10
3.0
4.0
Chapter 7: RV's & Probability Distributions
Section 7.4-7.5
1. What information about a probability distribution do the mean and standard deviation of a random variable
provide?
2. At a large university students have either a final exam or a final paper at the end of a course. The table
below lists the distribution of the number of final exams that students at the university will take, and their
associated probabilities. What are the mean and standard deviation of this distribution?
X
P(X)
0
0.05
1
0.25
2
0.40
3
0.30
Chapter 7, Review
Page 6 of 10
3. Suppose that the maximum daily temperature in Hacienda Heights, CA, for the month of December has a
mean of 17˚Celsius with a standard deviation of 3˚Celsius. Let F be the random variable maximum daily
9
temperature in degrees Fahrenheit. (Degrees F = C + 32 .)
5
a) What is the mean of F?
b) What is the standard deviation of F?
4. A State Dept. of Education is writing a state-wide math test, and by law must decide how many points will
count as a "failing score." The test consists of 50 True/False questions and 40 multiple choice questions
with 5 answer options. The total score (TS) will be equal to the number of true/false items correct plus
twice the number of multiple-choice items correct. A decision has been made to make the failing score the
score that a student would be expected to get if they randomly guessed on all the questions.
a) If a student is randomly guessing, the 50 True/False questions can be regarded as a binomial chance
experiment with probability of success equal to 0.50. If we define the random variable T = score from T/F
items, what are the mean and standard deviation of T for a random student who is guessing?
Chapter 7, Review
Page 7 of 10
b) If a student is randomly guessing, the 40 multiple choice questions can be regarded as a binomial chance
experiment with probability of success equal to 0.20. If we define the random variable M = score from MC
items, what are the mean and standard deviation of the M for a random student who is guessing?
c) The total score, TS, is a random variable formed by calculating T  2 M . What are the mean and standard
deviation of the random variable TS?
d) If a student is randomly guessing on the multiple choice part of the test, what is the probability that the first
multiple choice question correct is the 4th multiple choice question?
Chapter 7, Review
Page 8 of 10
Chapter 7: RV's & Probability Distributions
Section 7.6-7.7
1.
Determine the following areas under the standard normal (z) curve.
a) The area under the z curve to the left of 2.53
b) The area under the z curve to the left of –1.33
c) The area under the z curve to the right of 0.76
d) The area under the z curve to the right of –1.47
e) The area under the z curve between –1 and 3
f) The area under the z curve between –2.6 and –1.2
2. Let z denote a random variable having a standard normal distribution. Determine each of the following
probabilities.
a) P(z < 1.28)
b) P(z < - 1.05)
c) P(z > - 2.51)
d) P(- 1.30 < z < 1.54)
Chapter 7, Review
Page 9 of 10
3. A gasoline tank for a certain model car is designed to hold 12 gallons of gas. Suppose that the actual
capacity of the gas tank in cars of this type is well approximated by a normal distribution with mean 12.0
gallons and standard deviation 0.2 gallons. What is the probability that a randomly selected car of this
model will have a gas tank that holds at most 11.7 gallons?
4. The owners of the Burger Emporium are looking for new supplier of onions for their famous hamburgers. It
is important that the onion slice be roughly the same diameter as the hamburger patty. After careful
analysis, they determine that they can only use onions with diameters between 9 and 10 cm. Company A
provides onions with diameters that are approximately normally distributed with mean 10.3 cm and standard
deviation of 1.2 cm. Company B provides onions with diameters that are approximately normally
distributed with mean 10.6 cm and standard deviation of 0.9 cm. Which company provides the higher
proportion of usable onions? Justify your choice with an appropriate statistical argument.
Chapter 7, Review
Page 10 of 10