Download Prob Dist 11.3

Five-Minute Check (over Lesson 11–2)
CCSS
Then/Now
New Vocabulary
Example 1: Identify and Classify Random Variables
Key Concept: Probability Distribution
Example 2: Construct a Theoretical Probability Distribution
Example 3: Construct an Experimental Probability Distribution
Key Concept: Expected Value of a Discrete Random Variable
Example 4: Real-World Example: Expected Value
Key Concept: Standard Deviation of a Probability Distribution
Example 5: Real-World Example: Standard Deviation of a
Distribution
Over Lesson 11–2
WORK The number of hours worked in the last week by parttime employees at a fast food restaurant are 35, 20, 21, 32, 27,
12, 6, 16, 24, 19, 8, 18, 10, 28, 22, 30. Describe the center and
spread of the data using either the mean and standard
deviation or the five-number summary. Justify your choice.
A. The data are skewed. The range of hours worked is 6 to 35. The
median is 20.5 hours and the middle 50% is from 14 to 27.5 hours.
B. The data are symmetric. The range of hours worked is 6 to 35.
The median is 20.5 hours and the middle 50% is from 14 to 27.5
hours.
C. The data are skewed. The employees work an average of 20.5
hours per week with a standard deviation of about 8 hours.
D. The data are symmetric. The employees work an average of 20.5
hours per week with a standard deviation of about 8 hours.
Over Lesson 11–2
TICKETS The number of tickets sold for each event at a
concert venue are 319, 501, 366, 648, 413, 371, 468, 399,
711, 412, 679, 702, 575, 442, 324, 578. Describe the center
and spread of the data using either the mean and standard
deviation or the five-number summary. Justify your choice.
A. The data are symmetric. The range of ticket sales are from 319 to
711. The median is 455 and half of the ticket sales are between
385 and 613.
B. The data are skewed. The range of ticket sales are from 319 to
711. The median is 455 and half of the ticket sales are between
385 and 613.
C. The data are skewed. A mean of 494.25 tickets were sold with a
standard deviation of about 132 tickets.
D. The data are symmetric. A mean of 494.25 tickets were sold with a
standard deviation of about 132 tickets.
Over Lesson 11–2
BOWLING John and Justin compared their
bowling scores. Compare the distributions
using either the means and standard
deviations or the five-number summaries.
Justify your choice.
A. Neither distribution is skewed. The five number summary values for
John’s distribution are all greater than the corresponding values of
Justin’s distribution.
B. Justin’s scores are skewed. The five number summary values for John’s
distribution are all greater than the corresponding values of Justin’s
distribution.
C. John’s scores are skewed. The five number summary values for John’s
distribution are all greater than the corresponding values of Justin’s
distribution.
D. Their scores are symmetric. The five number summary values for John’s
distribution are all greater than the corresponding values of Justin’s
distribution.
Over Lesson 11–2
RUNNING Darren and Reggie compared
their times (in seconds) in the 40-meter
dash. Compare the distributions using
either the means and standard deviations
or the five-number summaries. Justify
your choice.
A. Both distributions are skewed. Reggie’s median time is lower than
Darren’s by two tenths of a second. The distributions have identical
ranges.
B. Both distributions are symmetric. Reggie’s median time is lower than
Darren’s by two tenths of a second. The distributions have identical
ranges.
C. Both distributions are skewed. Reggie’s mean time is lower than Darren’s
by almost four tenths of a second. The distributions have almost identical
standard deviations.
D. Both distributions are symmetric. Reggie’s mean time is lower than
Darren’s by almost four tenths of a second. The distributions have almost
identical standard deviations.
Over Lesson 11–2
SCORES Pam and Chee compared their quiz
scores in Science class for the year. The middle
half of the scores in Chee’s distribution are greater
than the bottom 75% of scores in Pam’s
distribution. The five number summary of Pam’s
distribution is 68, 72, 74, 76, 100. Which of the
following could be the five number summary of
Chee’s distribution?
A. 68, 72, 81, 88, 10
B. 70, 75, 80, 84, 100
C. 72, 77, 79, 82, 86
D. 71, 74, 82, 94, 100
Content Standards
S.MD.7 Analyze decisions and strategies using
probability concepts (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a
game).
Mathematical Practices
2 Reason abstractly and quantitatively.
You used statistics to describe symmetrical
and skewed distributions of data.
• Construct a probability distribution.
• Analyze a probability distribution and its
summary statistics.
• random variable
• experimental
probability
• discrete random variable
distribution
• continuous random
• Law of Large
variable
Numbers
• probability distribution
• expected value
• theoretical probability
distribution
Identify and Classify Random
Variables
A. Identify the random variable in the distribution,
and classify it as discrete or continuous. Explain
your reasoning.
the number of hits for the players of a baseball
team
Answer: The random variable X is the number of hits,
which is finite and countable, so X is
discrete.
Identify and Classify Random
Variables
B. Identify the random variable in the distribution,
and classify it as discrete or continuous. Explain
your reasoning.
the distances traveled by the tee shots in a golf
tournament
Answer: The random variable X is the distance
traveled, which can take on any value in a
certain range, so X is continuous.
Identify the random variable in the distribution, and
classify it as discrete or continuous. Explain your
reasoning.
the number of automobiles sold by an automaker in a
given week
A. Continuous; the number of automobiles sold
can take on any value.
B. Discrete; the number of automobiles sold is
finite and countable.
C. Discrete; the number of automobiles sold can
take on any value.
D. Continuous; the number of automobiles sold
is finite and countable.
Construct a Theoretical Probability Distribution
A. X represents the sum of two cards drawn from a
stack of cards numbered 1 through 8 with
replacement. Construct a relative-frequency table.
The theoretical probabilities associated with drawing two
cards from a stack numbered 1 through 8 can be
described using a relative-frequency table. When two
cards are drawn, 64 total outcomes are possible. To
determine the relative frequency, or theoretical
probability, of each outcome, divide the frequency by 64.
Construct a Theoretical Probability Distribution
Answer:
Construct a Theoretical Probability Distribution
B. X represents the sum of two cards drawn from a
stack of cards numbered 1 through 8 with
replacement. Graph the theoretical probability
distribution.
Construct a Theoretical Probability Distribution
Answer:
The graph shows the probability distribution
for the sum of the values on two dice X. The
bars are separated on the graph because
the distribution is discrete (no other values
of X are possible).
Each unique outcome of X is indicated on
the horizontal axis, and the probability of
each outcome occurring P(X) is indicated on
the vertical axis.
X represents the sum of two cards drawn from a
stack of cards numbered 1 through 5 with
replacement. Graph the theoretical probability
distribution.
A.
C.
B.
D.
Construct an Experimental Probability
Distribution
A. X represents the sum of two cards drawn from a
stack of cards numbered 1 through 8 with
replacement. Construct a relative frequency table for
64 trials.
Draw two cards 64 times or use a random number
generator to complete the simulation and create a
simulation tally sheet. Calculate the experimental
probability of each value by dividing its frequency by the
total number of trials, 64.
Construct an Experimental Probability
Distribution
Calculate the experimental probability of each value by
dividing its frequency by the total number of trials, 64.
Construct an Experimental Probability
Distribution
Answer:
Construct an Experimental Probability
Distribution
B. X represents the sum of two cards drawn from a
stack of cards numbered 1 through 8 with
replacement. Graph the experimental probability
distribution.
Answer: The graph shows the discrete probability
distribution for the sum of the values shown
on two dice X.
Expected Value
A tetrahedral die has four sides numbered 1, 2, 3,
and 4. Find the expected value of one roll of this die.
Each value has an equal chance of being rolled.
E(X) = Σ[X ● P(X)]
= 1(0.25) + 2(0.25) + 3(0.25) + 4(0.25)
= 0.25 + 0.50 + 0.75 + 1.00
= 2.5
Answer: 2.5
RAFFLES A group passed out 2000 raffle tickets.
The frequency table shows the number of winning
tickets for each prize. Find the expected value of
one ticket.
A. $0.88
B. $1.88
C. $2.75
D. $3.75
Standard Deviation of a
Distribution
RAFFLE At a raffle, 400 tickets are sold for $1 each.
One ticket wins $100, five tickets win $10, and ten
tickets win $5. Calculate the expected value and
standard deviation of the distribution of winnings for
a $1 ticket.
Make a frequency table. Of
the 400 tickets, 16 win a
prize, so 400 – 16 or 384 do
not. Divide the frequency by
400 to find the probability of
each prize. Each ticket costs
$1, so subtract 1 from each
prize value.
Standard Deviation of a
Distribution
E(X) = 0.0025(99) + 0.0125(9) + 0.0250(4) + 0.9600(–1)
= 0.2475 + 0.1125 + 0.1 + (–0.96)
= –0.5
Answer: The expected value of a ticket is –$0.50.
RAFFLES At a raffle, 500 tickets are sold for $5
each. One ticket wins $1000, two tickets win $500,
and ten tickets win $50. Calculate the expected
value and standard deviation of the distribution of
winnings for a $5 ticket.
A. $5; 3025
B. $5; 55
C. $0; 3050
D. $0; 55