Download S.8 - DPS ARE

Algebra 2: Interpreting Categorical and
Quantitative Data – S-ID
ELG.MA.HS.S.1: Summarize, represent, and
interpret data on a single count or measurement
variable.

S-ID.2 Use statistics appropriate to the shape
of the data distribution to compare center
(median, mean) and spread (interquartile
range, standard deviation) of two or more
different data sets.

S-ID.4 Use the mean and standard deviation
of a data set to fit it to a normal distribution
and to estimate population percentages.
Recognize that there are data sets for which
such a procedure is not appropriate. Use
calculators, spreadsheets, and tables to
estimate areas under the normal curve.
Conditional Probability and the Rules of
Probability – S-CP
ELG.MA.HS.S.6: Understand independence and
conditional probability and use them to interpret
data

S-CP.A.1 Describe events as subsets of a
sample space (the set of outcomes) using
characteristics (or categories) of the
outcomes, or as unions, intersections, or
complements of other events (“or,” “and,”
“not”).
Using Probability to Make Decisions – S-MD
ELG.MA.HS.S.8: Calculate expected values and use
them to solve problems.

S-MD.A.1 (+) Define a random variable for a
quantity of interest by assigning a numerical
value to each event in a sample space; graph the
corresponding probability distribution using the
same graphical displays as for data distributions.

S-MD.A.2 (+) Calculate the expected value of a
random variable; interpret it as the mean of the
probability distribution.

S-MD.A.3 (+)Develop a probability distribution
for a random variable defined for a sample
space in which theoretical probabilities can be
calculated; find the expected value. For
example, find the theoretical probability
distribution for the number of correct answers
obtained by guessing on all five questions of a
multiple-choice test where each question has
four choices, and find the expected grade under
various grading schemes.

S-MD.A.4 (+) Develop a probability distribution
for a random variable defined for a sample
space in which probabilities are assigned
empirically; find the expected value. For
example, find a current data distribution on the
number of TV sets per household in the United
States, and calculate the expected number of
sets per household. How many TV sets would
you expect to find in 100 randomly selected
households?
Students will demonstrate command of the ELG by:
 Defining a random variable for a quantity of interest by assigning a numerical value to each event in a sample space.
 Graphing the probability distribution of the sample space using the same graphical displays as for data distributions.
 Calculating the expected value of a random variable.



Interpreting the expected value of a random variable as the mean of the probability distribution.
Developing a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated and finding the expected value.
Developing a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically and finding the expected value.
Vocabulary:
 expected value
 mean
 probability distribution


random variable
sample space
Sample Assessment Questions:
1) Standard(s): S-MD.A.4
Source: Demana, F. D., Waits, B. K., Foley, G. D., Kennedy, D., & Bock, D.E. (2015) Precalculus: Graphical, Numerical, Algebraic. Pearson, 730.
Item Prompt:
A university widely known for its track and field program claims that 75% of its track athletes get degrees. A journalist investigates what happened to the 32 athletes who
began the program over a 6-year period that ended 7 years ago. Of these athletes, 17 have graduated and the remaining 15 are no longer attending any college. If the
university’s claim is true, the number of athletes who graduate among the 32 examined should have been governed by binomial probability with p = 0.75.
a. What is the probability that exactly 17 athletes should have graduated?
b. What is the probability that 17 or fewer athletes should have graduated?
c. If you were the journalist, what would you say in your story on the investigation?
Correct Answer(s):
a. Approximately 0.396
b. Approximately 0.596%
c. The university’s graduation rate claim seems to be exaggerated.
2) Standard(s): S-MD.A.3
Source: https://www.engageny.org/resource/precalculus-and-advanced-topics-module-5-topic-b-lesson-10/file/130096
Item Prompt:
A May 2000 Gallup poll found that 𝟑𝟖% of the people in a random sample of 𝟏, 𝟎𝟏𝟐 adult Americans said that they believe in ghosts. Suppose that three adults
will be randomly selected with replacement from the group that responded to this poll, and the number of adults (out of the three) who believe in ghosts will be
observed.
a. Develop a discrete probability distribution for the number of adults in the sample who believe in ghosts.
b. Calculate the probability that at least one adult, but at most two adults, in the sample believes in ghosts. Interpret this probability in context.
c. Out of the three randomly selected adults, how many would you expect to believe in ghosts? Interpret this expected value in context.
Correct Answer(s):
a.
Person 1
Person 2
Person 3
Calculation
Probability
NG
NG
NG
NG
NG
G
NG
G
NG
𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐
𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖
𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐
𝟎. 𝟐𝟑𝟖 𝟑𝟐𝟖
𝟎. 𝟏𝟒𝟔 𝟎𝟕𝟐
𝟎. 𝟏𝟒𝟔 𝟎𝟕𝟐
NG
G
G
𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖
𝟎. 𝟎𝟖𝟗 𝟓𝟐𝟖
G
NG
NG
𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐 × 𝟎. 𝟔𝟐
𝟎. 𝟏𝟒𝟔 𝟎𝟕𝟐
G
G
NG
𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐
𝟎. 𝟎𝟖𝟗 𝟓𝟐𝟖
G
NG
G
𝟎. 𝟑𝟖 × 𝟎. 𝟔𝟐 × 𝟎. 𝟑𝟖
𝟎. 𝟎𝟖𝟗 𝟓𝟐𝟖
G
G
G
𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖 × 𝟎. 𝟑𝟖
𝟎. 𝟎𝟓𝟒 𝟖𝟕𝟐
Note: G stands for “believing in ghosts,” and NG stands for “not believing in ghosts.”
Number of Adults Who Believe in Ghosts
Probability
𝟎
𝟏
𝟐
𝟑
𝟎. 𝟐𝟑𝟖 𝟑𝟐𝟖
𝟎. 𝟒𝟑𝟖 𝟐𝟏𝟔
𝟎. 𝟐𝟔𝟖 𝟓𝟖𝟒
𝟎. 𝟎𝟓𝟒 𝟖𝟕𝟐
b.
The probability that at least one adult, but at most two adults, believes in ghosts is 0.438 216+0.268 584=0.7068. If three adults were randomly selected and the
number of them believing in ghosts was recorded many times, the proportion that at least one, but at most two, adults believe in ghosts would be 0.7068.
c.
Out of three randomly selected adults, the expected number who believe in ghosts is as follows: 0 ∙ 0.238 328 + 1 ∙ 0.438 216 + 2 ∙ 0.268 584 + 3 ∙ 0.054 872 =
1.14 The long-run average number of adults in a sample of three who believe in ghosts is 1.14 adults.