Download Benchmark MACC.912.S-CP.2.6 Reporting Category Statistics and

1
Benchmark MACC.912.S-CP.2.6
Reporting Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
MACC.912.S-CP.2.6
Benchmark
Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A, and interpret the answer in terms of
the model.
Item Types
This benchmark will be assessed using MC and GR items.
Benchmark Clarification Find the conditional probability of an event given frequency data by
identifying the possible outcomes and determining their respective
probabilities.
Content Limits
Limit to real world contexts and age appropriate situations.
Stimulus Attribute
Tables and charts may be used.
Items may be set in either real-world or mathematical contexts.
Response Attributes
Gridded response items will have fractional answers.
The odds of an event must be expressed as a fraction.
Fractional answers must be in simplified form.
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SAMPLE ITEM
The state results for the FCAT 2.0 are shown in the table below. A student is considered
proficient if he obtains an Achievement Level of 3 or higher. What is the probability that an 8th
grade student was proficient on the math test in 2012? What are the odds that the student was
NOT proficient?
FCAT 2.0 Mathematics – Next Generation Sunshine State
Standards
FCAT 2.0
Percentage of Students
Number
Mean
by Achievement Level
Grade Year
of
Developmental 1
2
3
4
5
Students
Scale Score
3
2011 202,719
201
19 25 31 16
9
2012 203,207
202
18 24 30 18
10
4
2011 198,969
214
19 23 28 20
10
4
2012 193,802
215
18 22 27 20
12
5
2011 198,520
221
19 25 28 18
10
2012 199,844
222
19 24 27 18
11
6
2011 197,668
227
22 24 26 18
9
6
2012 199,076
227
23 25 25 18
10
7
2011 194,484
236
20 24 28 18
10
2012 198,277
236
20 24 27 18
10
8
2011 195,479
243
22 22 30 16
10
8
2012 194,346
243
22 21 30 16
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Answer, part 1:
30/100 + 16/100 + 11/100 = 57/100 or .57
Answer, part 2:
22/100 + 21/100 = 43/100
43/100 : 57/100 or 43 : 57
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Benchmark MACC.912.S-CP.2.7
Reporting
Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.2.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and
interpret the answer in terms of the model.
Also Assesses MACC.912.S-CP.1.2
or Assessed
by
Item Types
This benchmark will be assessed using MC items.
Content
Limit to real world contexts and age appropriate situations.
Limits
Benchmark
Clarification
Students will determine probable outcomes of events using the addition rule.
Stimulus
Attributes
Test items may include illustrations of the following: tables, lists, tree
diagrams, and other forms.
Response
Attributes
Sample Item
Responses may be in decimal or reduced fraction form.
If P(A) = 0.24 and P(B) = 0.52 and A and B are independent,
what is P(A or B)?
*
(a)
(b)
(c)
(d)
(e)
0.1248
0.28
0.6352
0.76
The answer cannot be determined from the information given.
Solution: 0.24 + 0.52 – (0.24)(0.52)
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Benchmark MACC.912.S-CP.2.8
Reporting
Category
Statistics and Probability
Domain
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.2.8
Also Assesses
or Assessed by
Item Types
Content Limits
MACC.912-CP.2.6/2.7, MACC.K12.MP.7.1
Benchmark
Clarification
Students will determine probabilities of single or multiple events using
probability addition and multiplication formulas.
Stimulus
Attributes
Items may be set in either real world or mathematical contexts. Items may
be a single event, multiple events, or a combination.
Response
Attributes
Responses will be in fraction or decimal form and should be reduced to
simplest form. Responses may be comprised of more than one answer. (find
probabilities of more than one item in a question)
Sample Item
Questions 1 and 2 relate to the following: An event A will occur with
probability 0.5. An event B will occur with probability 0.6. The probability
that both A and B will occur is 0.1.
Apply the general Multiplication Rule in a uniform probability model, P(A
and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the
model.
This benchmark will be assessed using MC, GR, and FR items.
Items may include multiple events or repeats of one event.
1. The conditional probability of A, given B
(a) is 0.5.
(b) is 0.3.
(c) is 0.2.
* (d) is 1/6.
(e) cannot be determined from the information given.
2. We may conclude that
(a) events A and B are independent.
(b) events A and B are disjoint.
* (c) either A or B always occurs.
(d) events A and B are complementary.
5
(e) none of the above is correct.
3.
SAMPLE ITEM
The Saffir-Simpson Hurricane Scale labels hurricanes with a 1-5 rating
based on their intensity. The overall probability that a hurricane is
rated 1-4 is 0.985. The probability of a hurricane hitting Florida is 0.32.
The probability that a hurricane is rated a 5 and misses Florida is 0.01.
Given this information, create and complete a two-way frequency table
and calculate the probability that a hurricane hits Florida given that it
has a category 5 rating.
Answer:
Rating of 1Rating of 5
4
Hits Florida
0.005
0.315
0.32
Misses
Florida
0.01
0.67
0.68
0.015
0.985
1
Defining
A1 = hits Florida
A2 = misses Florida
B = has rating of 5
Then
P(A1)P(B│A1)
P(A1│B) = ----------------------------------------P(A1)P(B│A1) + P(A2)P(B│A2)
(.32)(.005/.32)
P(A1│B) = ----------------------------------------(.32)(.005/.32) + (.68)(.01/.68)
Or
P(A1│B) = (.005/.015) = .333
= .333
Answer:
The probability that a hurricane hits Florida given that it has a
category 5 rating is .333
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Benchmark MACC.912.S-CP.1.1
Reporting
Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.1.1
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”).
Also Assesses MACC.K12.MP.4.1
or Assessed
by
Item Types
This benchmark will be assessed using MC and FR items.
Content
Limits
Limit to real world contexts and age appropriate situations.
Benchmark
Clarification
Students will determine all possible outcomes of events.
Stimulus
Attributes
Test items may include illustrations of the following: tables, lists, tree
diagrams, and other forms.
Response
Attributes
Sample Item
Responses may be in decimal or reduced fraction form.
3. What is the sample space of flipping a coin and rolling a die?
Answer: P(Coin and Dice ) = P(Coin U Dice) =
{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
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Benchmark MACC.912.S-CP.1.2
Reporting
Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.1.2
Understand that two events A and B are independent if the probability of A
and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent.
Also Assesses MACC.K12.MP.2.1
or Assessed
by
Item Types
This benchmark will be assessed using MC and FR items.
Content
Limits
Limit to real world contexts and age appropriate situations.
Benchmark
Clarification
Students will determine probable outcomes of events using the multiplication
rule.
Stimulus
Attributes
Test items may include illustrations of the following: tables, lists, tree
diagrams, and other forms.
Response
Attributes
Sample Item
Responses may be in decimal or reduced fraction form.
Toss two balanced coins. Let A = head on the first toss, and let B = both
tosses have the same outcome. Are events A and B independent? Explain
your reasoning clearly.
Solution: Yes, A and B are independent since P(A and B) is (.5)(.5)-0=0.25
and P(A)P(B)= (.5)(.5)=0.25
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SAMPLE ITEM 2 MC
Anthony attends a high school that has four periods in the school day. Last semester, his friend
took the same classes and recorded how often the teachers were absent. The data are shown in
the table below. There were 2 quarters in the semester and 45 teacher work days in each quarter.
Based on this data, what is the probability that all four of Anthony’s teachers will be absent on
any given day?
Period
Teacher Name
Subject
1
2
3
Mrs. Rodriguez
Ms. Williams
Mr. Keene
4
Mrs. Medina
Algebra
English
US History
Earth
Science
A)
B)
C)
D)
1st Quarter
Absences
2nd Quarter
Absences
5
1
0
4
2
10
3
1
1/60750
26/45
26/90
1/56782
Answer:
(9/90)(3/90)(10/90)(4/90) = 1/60750
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Benchmark MACC.912.S-CP.1.3
Reporting
Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.1.3
Understand the conditional probability of A given B as P(A and B)/P(B), and
interpret independence of A and B as saying that the conditional probability
of A given B is the same as the probability of A, and the conditional
probability of B given A is the same as the probability of B.
Also Assesses MACC.K12.MP.7.1
or Assessed
by
Item Types
This benchmark will be assessed using MC and FR items.
Content
Limits
Limit to real world contexts and age appropriate situations.
Benchmark
Clarification
Students will determine probable outcomes of events using the multiplication
rule.
Stimulus
Attributes
Test items may include illustrations of the following: tables, lists, tree
diagrams, and other forms.
Response
Attributes
Sample Item
Responses may be in decimal or reduced fraction form.
Use the conditional probability formula to demonstrate how the probability of
being dealt a queen given that the card is red, is the same as the probability of
being dealt a queen.
Answer: P(queen|red)= P (queen)·P(red) / P (red)
(4/52)·(26/52)/(26/52) = 4/52 = 1/13
P(queen) = 4/52 = 1/13
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Benchmark MACC.912.S-CP.1.4
Reporting
Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.1.4
Construct and interpret two-way frequency tables of data when two categories
are associated with each object being classified. Use the two-way table as a
sample space to decide if events are independent and to approximate
conditional probabilities. For example, collect data from a random sample of
students in your school on their favorite subject among math, science, and
English. Estimate the probability that a randomly selected student from your
school will favor science given that the student is in tenth grade. Do the same
for other subjects and compare the results.
Also Assesses MACC.K12.MP.2.1, MACC.K12.MP.4.1
or Assessed
by
Item Types
This benchmark will be assessed using MC and FR items.
Content
Limits
Limit to real world contexts and age appropriate situations.
Benchmark
Clarification
Students will determine probable outcomes of events using tables, lists, or
tree diagrams.
Stimulus
Attributes
Test items may include illustrations of the following: tables, lists, tree
diagrams, and other forms.
Response
Attributes
Sample Item
Responses may be in decimal or reduced fraction form.
Here are the counts (in thousands) of earned degrees in the United States in a
recent year, classified by level and by the sex of the degree recipient:
Female
Male
Total
Bachelor’s
616
529
1145
Master’s
194
171
365
Professional
30
44
74
Doctorate
16
26
42
Total
856
770
1626
1. If you choose a degree recipient at random, what is the probability that the
person you choose is a woman?
Answer : 856/1626
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2. What is the conditional probability that you choose a woman, given that
the person chosen received a professional degree?
Answer : 30/74
3. Are the events “choose a woman” and “choose a professional degree
recipient” independent? How do you know?
Answer: No if they were independent then P(woman) = P(woman|professional
degree), the answers to #1 and #2 would be the same.
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Benchmark MACC.912.S-CP.1.5
Reporting
Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.1.5
Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. For example,
compare the chance of having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.
Also Assesses MACC.912.S-CP.1.3 & MACC.912.S-CP.1.4
or Assessed
by
Item Types
This benchmark will be assessed using MC and FR items.
Content
Limits
Limit to real world contexts and age appropriate situations.
Benchmark
Clarification
Students will determine probable outcomes of events using the multiplication
rule.
Stimulus
Attributes
Test items may include illustrations of the following: tables, lists, tree
diagrams, and other forms.
Response
Attributes
Sample Item
Responses may be in decimal or reduced fraction form.
Researchers are interested in the relationship between cigarette smoking
and lung cancer. Suppose an adult male is randomly selected from a
particular population. Assume that the following table shows some
probabilities involving the compound event that the individual does or
does not smoke and the person is or is not diagnosed with cancer:
Event
Probability
smokes and gets cancer
0.08
smokes and does not get cancer
0.17
does not smoke and gets cancer
0.04
does not smoke and does not get cancer
0.71
Suppose further that the probability that the randomly selected individual
is a smoker is 0.25.
(a) Find the probability that the individual gets cancer, given that he is a
smoker.
13
Answer : 0.08/0.25 = 0.32
(b) Find the probability that the individual does not get cancer, given that
he is a smoker.
Answer : 0.17/0.25 = 0.68
(c) Find the probability that the individual gets cancer, given that he does
not smoke.
Answer : 0.04/0.75 = 0.053
(d) Find the probability that the individual does not get cancer, given that
he does not smoke.
Answer : 0.71/0.75 = 0.947
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Benchmark MACC.912.S-CP.2.9
Reporting
Category
Statistics and Probability
Standard
Conditional Probability & the Rules of Probability
Benchmark
Number
Benchmark
MACC.912.S-CP.2.9
Also Assesses
or Assessed
by
Item Types
Content
Limits
MACC.K12.MP.1.1, MACC.K12.MP.8.1
Benchmark
Clarification
Students will determine probable outcomes of events using the permutation
and combination formulas.
Stimulus
Attributes
Test items may include illustrations of the following: tables, lists, tree
diagrams, and other forms.
Response
Attributes
Sample Item
MC items of permutations should include answers using combination
formula.
1. Students are celebrating their academic success with ice cream
sundaes. There are 4 toppings available for their ice cream. Students may
choose 2 out of the 4 toppings for their ice cream. This includes an option for
a double helping of one topping. How many two-topping combinations can
students choose?
Use permutations and combinations to compute probabilities of compound
events and solve problems.
This benchmark will be assessed using MC and FR items.
Items shall be limited to no more than five different sets. Limit to real world
contexts and age appropriate situations.
Answer : 4C2 + 4= 6 + 4 =10
2. The school is going to hold an Algebra competition. Each teacher is to
randomly select 2 students to represent their class. If Ms. Brown’s class has
10 boys and 14 girls, what is the probability that one boy and one girl will be
chosen to represent her class?
Answer : (10C1 · 14C1) / 24C2 = (10· 14) / 276 = 35/69
15
Benchmark MACC.912.S-IC.1.1
Reporting
Category
Standard
Benchmark
Number
Benchmark
Probability and Statistics
Benchmark
Clarification
Students will calculate, summarize, and interpret data using measures of
center including mean and median and measures of spread including range,
standard deviation, and variance. Students will use these measures to make
comparisons and draw conclusions about data sets.
Also Assesses
or Assessed by
Item Types
MACC.K12.MP.5.1, MACC.K12.MP.6.1
Content Limits
Items may include calculations for mean, median, variance, and standard
deviation. Items may include the choice of which measure of center is the
best representation of a data set.
Benchmark
Clarification
Students will calculate, summarize, and interpret data using measures of
center and spread.
Stimulus
Attributes
Response
Attributes
Items must be set in mathematical or real-world context.
Sample Items
Sample Item 1
MC
The average grade point average (GPA) for 25 top-ranked students at a
local college is listed below. (Use the rounding rule for the mean).
3.80 3.77 3.70 3.74 3.70
3.86 3.76 3.68 3.67 3.57
3.83 3.70 3.80 3.73 3.67
3.78 3.75 3.73 3.65 3.66
3.75 3.64 3.78 3.73 3.64
Find the mean of the GPAs.
Sample Response 1
A.
3.7236
B.
3.764
Making Inferences & Justifying Conclusions
MACC.912.S-IC.1.1
Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
This benchmark will be assessed using FR, GR, and MC items.
Responses will be rational numbers.
16
*C.
3.724
D.
3.887
Sample Response 1 C
Item Context:
Mathematics (Statistics)
Sample Item 2
MC
The average grade point average (GPA) for 25 top-ranked students at a
local college is listed below. (Use the rounding rule for the mean).
3.80 3.77 3.70 3.74 3.70
3.86 3.76 3.68 3.67 3.57
3.83 3.70 3.80 3.73 3.67
3.78 3.75 3.73 3.65 3.66
3.75 3.64 3.78 3.73 3.64
Find the median of the GPAs.
Sample Response 2
A.
3.7
*B.
3.73
C.
3.74
D.
3.75
Sample Response 2 B
Item Context:
Mathematics (Statistics)
17
Sample Item
Sample Item 3 FR
Summer income for 14 high school students is listed below:
1,208 2,400
1,337 3,020
1,055
456
810 1,765
1,208
987
2,343 1,654
734 1,356
Find the standard deviation. (Use the rounding rule for standard deviation).
7
Sample Response 3
Item Context
1
9
.
2
719.2
Mathematics (Statistics)
Sample Item 4 FR
Summer income for 14 high school students is listed below:
1,208 2,400
1,337 3,020
1,055
456
810 1,765
1,208
987
2,343 1,654
734 1,356
Find the variance.
5
1
7
2
4
9
Sample Response 4 517249
Item Context
Mathematics (Statistics)
18
Benchmark MACC.912.S-IC.1.2
* Teacher may access in the classroom.
Reporting
Category
Standard
Benchmark
Number
Benchmark
Probability and Statistics
Also Assesses
or Assessed by
Item Types
LACC.9.10.WHST.1.1, LACC.9.10.WHST.2.4
Content Limits
Items may include calculations for mean, median, variance, and standard
deviation.
Benchmark
Clarification
Students will calculate, summarize, and interpret data using measures of
center and spread. Students will use these measures to make comparisons
and draw conclusions about statistics and probability models.
Stimulus
Attributes
Response
Attributes
Items must be set in mathematical or real-world context.
Making Inferences & Justifying Conclusions
MACC.912.S-IC.1.2
Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says a
spinning coin falls heads up with probability 0.5. Would a result of 5 tails in
a row cause you to question the model?
This benchmark will be assessed using FR or MC items.
Responses may be written understanding of inferences conditions.
* Teacher may access in the classroom.
Sample Item
Suppose you have a population in which 60% of the people approve of gambling.
You want to take many samples of size 10 from this population to observe how the sample proportion
who approve of gambling varies in repeated samples.
*1. Describe the design of a simulation using the partial random digits table below to estimate the sample
proportion who approve of gambling. Then carry out five trials of your simulation.
19
6 0 0 9
1 9 3 6 5
1 5 4 1 2
3 9 6 3 8
8 5 4 5 3
4 6 8 1 6
3 8 4 4 8
4 8 7 8 9
1 8 3 3 8
2 4 6 9 7
3 9 3 6 4
4 2 0 0 6
8 2 7 3 9
5 7 8 9 0
2 0 8 0 7
4 7 5 1 1
8 1 6 7 6
5 5 3 0 0
6 0 9 4 0
7 2 0 2 4
1 7 8 6 8
2 4 9 4 3
6 1 7 9 0
9 0 6 5 6
6 8 4 1 7
3 5 0 1 3
1 5 5 2 9
7 2 7 6 5
8 5 0 8 9
5 7 0 6 7
*2. The sampling distribution is the distribution of all possible simple random samples of size 10 from
this population. What is the mean of this distribution?
Answer: p = 0.6 is the mean of the distribution, to find the sample mean average the results from
question 1.
Free Response Item
3. If you used samples of size 20 instead of size 10, which sampling distribution would give you a better
estimate of the true proportion (of people who approve of gambling)? Explain your answer.
Answer: Samples of size 20 would give better estimates because of decreased variability.
20
Benchmark MACC.912.S-IC.2.3
Reporting
Category
Statistics and Probability
Standard
Benchmark
Number
Benchmark
Making Inferences & Justifying Conclusions
MACC.912.S-IC.2.3
Also Assesses
MACC.K12.MP.3.1
Item Types
This benchmark will be assessed using MC and FR items.
Content
Limits
Items may include research questions from surveys, observational studies, or
experiments. Items may include sampling methods such as simple random,
systematic, stratified, or cluster.
Benchmark
Clarification
Students will use and compare surveys, experiments, and observational
studies and decide which questions each is designed to answer.
Student will use qualitative and quantitative measures to collect data
Students will choose the type of sampling from a variety of situations as well
as understand the reasoning behind random sampling.
Students will identify sources of bias, including sampling and non-sampling
errors.
Stimulus
Attributes
Response
Attributes
Sample Item
Items must be set in real-world context.
Recognize the purposes of and differences among sample surveys,
experiments, and observational studies; explain how randomization relates to
each.
Answers need to be situations that are possible.
1. After a class receives the same lesson from the same teacher the class is
divided randomly into two equal sized groups. One group is given a multiple
choice test and the other group is given an essay test. Then the average of the
test scores from each group are compared.
What is the technique for gathering data in this situation?
A. survey
B. observational study
*C. experiment
D. survey/experiment
21
2. Group businesses in Orlando, Fl according to type: medical, shipping,
retail, manufacturing, financial, construction, restaurant, hotel, tourism, other.
Then select a random sample of 10 businesses from each business type to
survey.
What type of sampling is being used in this situation?
A. simple random sampling
B. cluster sampling
C. systematic sampling
*D. stratified sampling
3. Students in grades 9-12 at an area high school were asked whether or not
they should have tougher classes to help improve their testing performance on
state assessments?
What form of bias was used in this situation?
A. type of question
B. language
*C. sample
D. misleading
22
Benchmark MACC.912.S-IC.2.4
Reporting
Category
Standard
Benchmark
Number
Benchmark
Probability and Statistics
Also Assesses
or Assessed by
Item Types
MACC.K12.MP.5.1, MACC.K12.MP.6.1
Content Limits
Items may include confidence level and/or confidence interval. Items may
include intervals for the mean when the population standard deviation is
known and when the population standard deviation is not known. Items may
include confidence intervals for proportions also. Items may include finding
sample size needed for specific results.
Benchmark
Clarification
Students will create, explain and interpret confidence intervals of a mean or
proportion by understanding point estimate, confidence levels and sample
size when the population standard deviation is known and when the
population standard deviation is not known. Students will also interpret and
apply the maximum error of the estimate ("margin of error").
Stimulus
Attributes
Response
Attributes
Items must be set in mathematical or real-world context.
Making Inferences & Justifying Conclusions
MACC.912.S-IC.2.4
Use data from a sample survey to estimate a population mean or proportion;
develop a margin of error through the use of simulation models for random
sampling.
This benchmark will be assessed using GR and MC items.
Answers will include confidence levels in percentages, confidence intervals
with a mean or proportion estimate or sample sizes in whole numbers.
Sample Items
1. When people smoke, the nicotine they absorb is converted to cotinine,
which can be measured. A sample of 40 smokers has a mean cotinine level
of 172.5. Assuming the population standard deviation is known to be 119.5,
what is the 90% confidence interval estimate of the mean cotinine level of all
smokers?
A. 74.64 < 𝑥̅ < 164.36
*B. 141.42 < 𝑥̅ < 203.58
C. 166.49 < 𝑥̅ < 178.51
D. 141.42 < 𝑥̅ < 178.51
23
24
Gridded Response
2. The music industry recently has adjusted to the increasing number of songs being downloaded
instead of being purchased on CDs. So, it has become important to estimate the proportion of
songs being currently downloaded. If we want to be 95% confident that the sample percentage is
within 1% point of the true population percentage of songs obtained by downloading, how many
randomly selected song purchases must be surveyed to determine the percentage downloaded?
Sample Response
9
6
0
4
25
Benchmarks: MACC.912.S-IC.2.5, & MACC.912.S-IC.2.6
Body of
Knowledge
Standard
Statistics and Probability
Benchmark
Number
MACC.912.S-IC.2.5 & MACC.912.S-IC.2.6
Benchmark
MACC.912.S-IC.2.5: Use data from a randomized experiment to compare two
treatments; use simulations to decide if differences between parameters are
significant.
Making Inferences & Justifying Conclusions
MACC.912.S-IC.2.6: Evaluate reports based on data.
Also assesses or
assessed by
LACC.910.WHST.1.1, LACC.910.WHST.2.4, LACC.910.WHST.3.9, and
MACC.K12.MP.3.1, MACC.912.S-IC.1.1
Item Types
This benchmark will be assessed using and MC, GR and FR items.
Benchmark
Clarification
Students will understand, create and analyze the null hypothesis and the
alternative hypothesis (basics of hypothesis testing).
Students will understand the p-value is the probability of getting a sample
statistic in the direction of the alternative hypothesis when the null
hypothesis is true. Students will understand p-value rules for decisions to
reject or not to reject the null hypothesis.
Students will decide whether a Type I error (occurs when you reject the
null hypothesis when it is true) and a Type II error (occurs if you do not
reject the null hypothesis when it is false) have occurred in the outcome of
the hypothesis testing.
Content Limits
Items may include writing a null and alternative hypothesis for data or
interpreting a given null and alternative hypothesis.
Items may include stating the hypothesis, identifying the claim, computing
the test value, finding the p-value, making a decision on given information
and/or summarizing the results.
Stimulus
Attribute
Response
Attributes
Items must be set in mathematical or real-world context.
Items may be writing the null and alternative hypothesis or analyzing the
given null and alternative hypothesis.
Answers may include the p-value and/or a decision to be made based on
the p-value.
26
Sample Item 1
FR
The average undergraduate cost for tuition, fees, room, and board for all accredited universities
was $26,025. A random sample in 2009 of 40 of these accredited universities indicated that the
mean tuition, fees, room, and board for the sample was $27,690 with a population standard
deviation of $5492. What is the null and alternative hypothesis for the conjecture?
A.
B.
C.
D.

*
H˳: µ= $26,025
H˳: µ < $26,025
H˳: µ =$26,025
H˳:µ = $26,025
Sample Item 2
and
and
and
and
Ң: µ > $26,025
Ң: µ = $26,025
Ң: µ < $26,025
Ң: µ ≠ $26,025
FR
Using the statistics from sample item 1 and a 0.05 level of significance, what is the p-value
of this sample?
Sample Response
0.
0
2
7
4
Sample Item 3
MC
The average undergraduate cost for tuition, fees, room, and board for all accredited universities
was $26,025. A random sample in 2009 of 40 of these accredited universities indicated that the
mean tuition, fees, room, and board for the sample was $27,690 with a population standard
deviation of $5492. At a 0.05 level of significance, which of these statements is the correct
conclusion for the hypothesis test using the p-value?
*
A.

B.
C.
D.
Item Context
Reject the null hypothesis with p-value of 0.0274 with a significance level of
0.05.
Do not reject the null hypothesis with p-value of 0.0274 with a significance
level of 0.05.
Except the null hypothesis with p-value of 0.0274 with a significance level of
0.05.
Not enough information to formulate a conclusion.
Probability & Statistics
27
Sample Item 4
MC
The hypothesis of a sample gives the following conclusion for proportion of car crashes occurring
less than a mile from home:
The proportion of car crashes occurring less than a mile from home is 0.23.
What are the Type I and Type II errors which could happen with these results?
A.
*B.
C.
D.
Item Context
Type I error: Fail to reject the claim that the proportion of car crashes
occurring less than a mile from home is 0.23 when the proportion is actually
different from 0.23.
Type II error: Reject the claim that the proportion of car crashes occurring
less than a mile from home is 0.23 when that proportion is actually 0.23.
Type I error: Reject the claim that the proportion of car crashes occurring
less than a mile from home is 0.23 when that proportion is actually 0.23.
Type II error: Fail to reject the claim that the proportion of car crashes
occurring less than a mile from home is 0.23 when the proportion is actually
different from 0.23.
Type I error: Reject the claim that the proportion of car crashes occurring
less than a mile from home is 0.23 when that proportion is actually different
from 0.23.
Type II error: Fail to reject the claim that the proportion of car crashes
occurring less than a mile from home is 0.23 when the proportion is actually
0.23.
Type I error: Fail to reject the claim that the proportion of car crashes
occurring less than a mile from home is 0.23 when the proportion is actually
0.23.
Type II error: Reject the claim that the proportion of car crashes occurring
less than a mile from home is 0.23 when that proportion is actually different
from 0.23.
Probability & Statistics
28
Benchmark MACC.912.S-ID.1.1
Reporting
Category
Standard
Statistics and Probability
Benchmark
Number
Benchmark
MACC.912.S-ID.1.1
Also
Assesses or
Assessed by
Item Types
MACC.K12.MP.4.1
Content
Limits
Benchmark
Clarificatio
n
Stimulus
Attributes
Response
Attributes
Sample
Item
Items will include raw data.
Interpreting Categorical & Quantitative Data
Represent data with plots on the real number line (dot plots, histograms, and box
plots).
This benchmark will be assessed using FR or MC items.
Students will create a data distribution in the form of: dot plots, histograms, or
box plots.
Data must be rational numbers
Graphs must be appropriately labeled and draw to scale.
Here are the season home run statistics of two major league baseball players I in
increasing order. Draw parallel box plots to compare the two players.
Here are numbers for the first player:
16
19
24
25
25
25
34
37
37
40
42
45
45
46
33
33
33
46
49
73
32
34
45
47
Here are the numbers for the second player:
13
20
24
26
27
29
30
38
39
39
40
40
44
44
44
44
Draw parallel box plots for the two players:
HRcodes
1
*
2
HomeRuns
15
30
45
60
75
34
34
29
Benchmark MACC.912.S-ID.1.2
Reporting
Category
Standard
Statistics and Probability
Benchmark
Number
Benchmark
MACC.912.S-ID.1.2
Also
Assesses or
Assessed by
Item Types
MACC.912.S-ID.1.1, MACC.K12.MP.5.1
Content
Limits
Benchmark
Clarification
Items may include calculating measures of center and spread.
Stimulus
Attributes
Response
Attributes
Sample Item
Items must be set in real-world context.
Interpreting Categorical & Quantitative Data
Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) or
five number summaries of two or more different data sets.
This benchmark will be assessed using FR and MC items.
Students will calculate measures of center and spread of multiple data sets and
compare the results.
Answers may include mean, median, IQR, and standard deviation.
Sample Item 1
MC
The average grade point average (GPA) for 25 top-ranked students at a
local college is listed below.
3.80 3.77 3.70 3.74 3.70
3.86 3.76 3.68 3.67 3.57
3.83 3.70 3.80 3.73 3.67
3.78 3.75 3.73 3.65 3.66
3.75 3.64 3.78 3.73 3.64
Find the five-number summary for the GPAs.
Sample Response 1
A.
3.64, 3.675, 3.73, 3.775, 3.86
*B.
3.57, 3.67, 3.73, 3.775, 3.86
C.
3.64, 3.67, 3.73, 3.775, 3.80
D.
3.57, 3.67, 3.73, 3.775, 3.9
Sample Response 1 B
Item Context:
Mathematics (Statistics)
30
2. Use graphs and numerical summaries to describe how the following three
data sets are similar and how they are different.
A: 5, 7, 9, 11, 13, 15, 17
B: 5, 6, 7, 11, 15, 16, 17
C: 5, 5, 5, 11, 17, 17, 17
Answer:
Similar- Each set has the same mean of 11 and the same median of 11 all
three sets are symmetric. All three sets have the same minimum value of 5
and maximum value of 17.
Different – They all have different standard deviations (spread)
A. 4.32, B. 5.07, C. 6.0.
3. Sample Item
MC
Which statement describes the histogram appropriately?
*A. The histogram is skewed to the right (positively skewed).
B. The histogram is skewed to the left (negatively skewed).
C. The histogram is a normal distribution.
D. The histogram is bimodal.
Item Context: Mathematics (Statistics)
31
Benchmark MACC.912.S-ID.1.3
Reporting
Category
Standard
Statistics and Probability
Benchmark
Number
Benchmark
MACC.912.S-ID.1.3
Also
Assesses or
Assessed by
Item Types
Content
Limits
MACC.K12.MP.4.1, MACC.912.S-ID.1.2
Benchmark
Clarificatio
n
Stimulus
Attributes
Response
Attributes
Students will identify outliers, the possible reasons for them and what effect
they have on the mean, median, mode and range of data.
Sample
Item
Summer income for 14 high school students is listed below:
1,208 2,400
1,337 3,020
1,055
456
810 1,765
1,208
987
2,343 1,654
734 1,356
Interpreting Categorical & Quantitative Data
Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme data points (outliers).
This benchmark will be assessed using FR or MC items.
Items may include identifying outliers from a set of data in a graph such as a
histogram or box-and-whisker plot. Items may also include what effect adding a
higher or lower value have on a particular mean, median, mode or range.
Items must be set in real-world context.
Answers will include numerical outliers, reasons for outliers, or effect of
specific outlier on mean, median, mode or range of data.
Which of these statements is true?
A. 456 is an outlier.
*B. 3020 is an outlier.
C. 456 and 3020 are outliers.
D. There are no outliers.
32
Benchmark MACC.912.S-ID.1.4
Reporting
Category
Standard
Statistics and Probability
Benchmark
Number
Benchmark
MACC.912.S-ID.1.4
Interpreting Categorical & Quantitative Data
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.
Also
MACC.K12.MP.6.1
Assesses or
Assessed by
Item Types This benchmark will be assessed using FR, GR, and MC items.
Content
Limits
Items may include finding area under a curve between z-scores, finding zscores with given probability or finding specific data values with given
probability. Items may include students analyzing graphs with z-sores, data
values or probabilities.
Benchmark
Clarificatio
n
Students will identify the properties of a normal distribution, find area under the
standard normal distribution, find probabilities of a variable under a normal
distribution, and find specific data values for given percentages using the
standard normal distribution.
Stimulus
Attributes
Response
Attributes
Items must be set in real-world context.
Sample
Item
Sample Item 1
Items may be in decimal or percentage form. Items should specify whether the
answer expected is in decimal or percentage form. Z-scores may be negative.
FR
What is the probability (in decimal form) with the given information
using the standard normal distribution?
P (-2.46 < z < 1.74)
Sample Response
0
.
9
5
2
1
33
Sample Item 2
MC
For men aged between 18 and 24 years, serum cholesterol levels (in
mg/100mL) have a mean of 178.1 and a standard deviation of 40.7 (based
in data from the National Health Survey). What is the z-score
corresponding to a male, aged 18-24 years, who has a serum cholesterol
level of 259.0 mg/100mL?
Sample Response 2
A.
-1.99
*B.
1.99
C.
6.36
D.
80.9
Sample Response 2 B
Item Context:
Mathematics (Statistics)
Sample Item 3
FR
Women’s heights have a mean of 63.6 inches and a standard deviation of
2.5 inches. What is the z-score corresponding to a woman with a height of
70 inches?
Sample Response 3
2
.
5
6
Sample Response 2.56
Item Context:
Mathematics (Statistics)
34
Benchmark MACC.912.S-ID.2.5
Reporting
Category
Standard
Statistics and Probability
Benchmark
Number
Benchmark
MACC.912.S-ID.2.5
Interpreting Categorical & Quantitative Data
Summarize categorical data for two categories in two-way frequency tables.
Interpret relative frequencies in the context of the data (including joint,
marginal, and conditional relative frequencies). Recognize possible associations
and trends in the data.
Also
MACC.K12.MP.4.1, LACC.910WHST.3.9
Assesses or
Assessed by
Item Types This benchmark will be assessed using FR and MC items.
Content
Limits
Benchmark
Clarificatio
n
Stimulus
Attributes
Response
Attributes
Items will be presented in two-way frequency tables.
Sample
Item
In 2006, an electronic replay system debuted in both men’s and women’s
professional tennis. Each player is allowed two unsuccessful challenges per set.
Here are some data on the results of challenges made during the first few
months of the new system.
Students will calculate and interpret frequencies found in two-way tables.
Items must be set in real-world context.
Answers may be in decimal or percentage form. Answer may require written
statements.
Men
Women
Successful
201
126
Unsuccessful
288
224
1. Calculate the marginal distribution (in percents) for each of the two
variables.
Answer: Gender- Men 489/839 = 58.28%, Women 350/839 =41.72%
Challenge results- Successful : 327/839 = 38.97%, Unsuccessful- 512/839 =
61.03%
2. Write a sentence describing what each marginal distribution tells you.
A higher percentage of men make challenges than women. 61% of all
challenges are unsuccessful.
35
3. Write a few sentences describing the relationship between gender and
challenge results.
Of the men who made challenges about 41% were successful. Only 36% of
the women who made challenges were successful.
36
Benchmark MA.912.S-ID.2.6 & MACC.912.S-ID.3.7
Reporting Category
Standard
Benchmark
Probability and Statistics
MACC.912.S-ID.2.6 & MACC.912.S-ID.3.7
MACC.912.S-ID.2.6
1. Represent data on two quantitative variables on a scatter plot, and
describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve
problems in the context of the data. Use given functions or choose a
function suggested by the context. Emphasize linear, quadratic, and
exponential models.
b. Informally assess the fit of a function by plotting and analyzing
residuals.
c. Fit a linear function for a scatter plot that suggests a linear
association.
MACC.912.S-ID.3.7
2. Interpret the slope (rate of change) and the intercept (constant
term) of a linear model in the context of the data.
Also Asses
MACC.K12.MP.4.1, MACC.K12.MP.5.1, MACC.K12.MP.8.1
Item Types
This benchmark will be assessed using MC and FR items.
Benchmark
Clarification
Students will graph data on a scatter plot, describe the association,
and interpret the slope.
Content Limits
Stimulus Attributes
Data points limited to one, two, or three digits
Tables and charts may be used.
Items may be set in either real-world or mathematical contexts.
Response Attributes
Equations may be presented in standard or slope-intercept form.
* Teacher may assess in the classroom.
37
SAMPLE ITEM
Mrs. Santiago would like to know how well her Algebra students will perform on the state’s
End-of-Course Assessment. She has decided to gather some data from the previous year in order
to determine a least squares regression line. She plans to use the students’ scores on her
classroom test to predict how well they will do on the End-of-Course Assessment. Given the
data in the table below
I. Graph the data on a scatter plot and describe how the variables are related.
Answer: *Teacher may assess in the classroom.
II. Find the equation of the least squares regression line.
III. Interpret the slope (rate of change) and the intercept (constant term) of this linear model in the context
of the data.
Classroom Test
Score
90
80
75
40
80
85
II.
A)
B)
C)
D)
y = .11x + 14.73
y = 14.73x + .11
y = 271.17x + 1.98
y = 1.98x + 271.17
Answer:
D)
Solution:
Algebra EOC Scale
Score
450
425
425
350
430
440
38
(n∑xy)-(∑x∑y)
m = ---------------------- =
(n∑x2)-(∑x)2
b=
(6 * 192,175) – (450 * 2520)
---------------------------------------- = 1.98
(6 * 35350) – (450)2
(∑y∑x2)-(∑x∑xy)
(2520 * 35350) – (450 * 192,175)
------------------------ = ---------------------------------------------- = 271.17
(n-∑x2)-(∑x)2
(6 * 35350) – (450)2
y = mx + b = 1.98x + 271.17
III.
Slope: The Algebra EOC Scale Score will rise 1.98 points for every one point increase on
the classroom test score.
Y-Intercept: When the classroom test score is 0, the Algebra EOC score will be 271.17
39
Reporting
Category
Standard
Benchmark
Number
Benchmark
Benchmark MACC.912.S-ID.3.8 & MACC.912.S-ID.3.9
Statistics and Probability
Interpreting Categorical & Quantitative Data
MACC.912.S-ID.3.8
MACC.912.S-ID.3.8: Compute (using technology) and interpret the correlation
coefficient of a linear fit.
MACC.912.S-ID.3.9: Distinguish between correlation and causation.
Build on students’ work with linear relationships in eighth grade and introduce
the correlation coefficient. The focus here is on the computation and
interpretation of the correlation coefficient as a measure of how well the data fit
the relationship. The important distinction between a statistical relationship and
a cause-and-effect relationship arises in S.ID.9.
Also
MACC.K12.MP.2.1
Assesses or
Assessed by
Item Types This benchmark will be assessed using FR, GR, or MC items.
Content
Limits
Items may include calculating the correlation coefficient of a set of paired data
and the interpretation of the relationship between the two variables.
Benchmark
Clarificatio
n
Students will calculate the correlation coefficient of a set of paired data (x,y) to
determine the strength and direction of the relationship between the variables.
Student will also use scatter plots to aid their interpretation of the data.
Stimulus
Attributes
Response
Attributes
Sample
Item
Items must be set in real-world context.
Answers will include the correlation coefficient and/or the interpretation of a
given correlation coefficient.
Sample Item 1
FR
The table below lists the numbers of murders and the population sizes (in
hundreds of thousands) for large cities in America during a recent year.
What is the correlation coefficient of the data?
Murders
Populatio
n
25
8
4
26
4
6
40
2
9
25
3
6
11
1
3
64
8
29
28
8
15
65
4
38
25
6
20
6
0
6
59
0
81
40
Sample Response 1
0
.
7
2
7
Sample Response 1
0.727
There is a strong positive relationship between the number of
murders and the population size of large American cities.
41
Benchmark MACC.912.S-MD.1.1
*Teacher may assess in the classroom.
Reporting Category
Statistics and Probability
Standard
Using Probability to Make Decisions
Benchmark Number
MACC.912.S-MD.1.1
Benchmark
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.
Also assesses or
assessed by
Item Types
MACC.K12.MP.4.1
Benchmark
Clarification
Students will graph a random variable distribution on a scatter plot.
Content Limits
Items may include probability distributions or raw data. Items may include
discrete random variables, discrete uniform variables, binomial or
exponential data.
Stimulus Attribute
Items must be set in mathematical or real-world context.
Response Attributes
Items may be in fraction or decimal form.
*Teacher may assess in the classroom.
This benchmark will be assessed using FR or MC items
42
*Teacher may assess in the classroom.
Sample Item 1
FR
The probabilities that a randomly selected customer purchases 1, 2, 3, 4, or 5 items at a
convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively.
(a) Identify the random variable of interest. X = ________. Then construct a probability
distribution (table), and draw a probability distribution histogram.
Solution:
X= number of items purchased.
43
Benchmark MACC.912.S-MD.1.2
Reporting Category
Statistics and Probability
Standard
Using Probability to Make Decisions
Benchmark Number
MACC.912.S-MD.1.2
Benchmark
Calculate the expected value of a random variable; interpret it as the mean of
the probability distribution.
Also assesses or
assessed by
Item Types
MACC.K12.MP.5.1
Benchmark
Clarification
Students will use the formula to calculate the expected value (mean) and
variance of a random variable.
Content Limits
Items may include probability distributions or raw data. Items may include
discrete random variables, discrete uniform variables, binomial or
exponential data.
Stimulus Attribute
Items must be set in mathematical or real-world context.
Response Attributes
Items may be in fraction or decimal form.
This benchmark will be assessed using MC and GR items
Sample Item 1
MC
X
P(X)
4
0.4
5
0.3
A) What is the mean of the Probability Distribution?
Solution: 4(0.4) + 5(0.3) +6(0.1) + 8(0.15) + 10(0.05) = 5.4
6
0.1
8
0.15
10
0.05
44
What is the variance of the Probability Distribution?
*
A.
B.
C.
D.
0.35
1.71
2.94
5.4
Solution: (4-5.4)²·(0.4) + (5-5.4)²·(0.3) + (6-5.4)²·(0.1) + (8-5.4)²·(0.15) + (10-5.4)²·(0.05) = 2.94
Item Context
Probability and Statistics
45
Benchmark MACC.912.S-MD.1.3
*Teacher will assess in the classroom.
Reporting Category
Statistics and Probability
Standard
Using Probability to Make Decisions
Benchmark
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.
Also assesses or
assessed by
Item Types
MACC.K12.MP.4.1
Benchmark
Clarification
Students will use the formula to calculate the expected value (mean) of a
random variable.
Content Limits
Items may include probability distributions or raw data. Items may include
discrete random variables, discrete uniform variables, binomial or
exponential data.
Stimulus Attribute
Items must be set in mathematical or real-world context.
Response Attributes
Items may be in fraction or decimal form.
Sample Item
*Teacher will assess in the classroom.
This benchmark will be assessed using FR or MC items
46
Benchmark MACC.912.S-MD.1.4
*Teacher will assess in the classroom.
Reporting Category
Statistics and Probability
Standard
Using Probability to Make Decisions
Benchmark
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?
Also assesses or
assessed by
Item Types
MACC.K12.MP.4.1
Benchmark
Clarification
Students will use the formula to calculate the expected value (mean) of a
random variable.
Content Limits
Items may include probability distributions or raw data. Items may include
discrete random variables, discrete uniform variables, binomial or
exponential data.
Stimulus Attribute
Items must be set in mathematical or real-world context.
Response Attributes
Items may be in fraction or decimal form.
Sample Item
*Teacher will assess in the classroom.
This benchmark will be assessed using FR or MC items
47
Benchmark MACC.912.S-MD.2.5 & MACC.912.S-MD.2.6
Reporting Category
Statistics and Probability
Standard
Using Probability to Make Decisions
Benchmark
MACC.912.S-MD.2.5 : Weigh the possible outcomes of a decision by
assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the
expected winnings from a state lottery ticket or a game at a fast-food
restaurant.
b. Evaluate and compare strategies on the basis of expected values. For
example, compare a high-deductible versus a low-deductible automobile
insurance policy using various, but reasonable, chances of having a minor
or a major accident.
MACC.912.S-MD.2.6: Use probabilities to make fair decisions (e.g.,
drawing by lots, using a random number generator).
Also assesses or
assessed by
Item Types
MACC.K12.MP.4.1
Benchmark
Clarification
Students will use the formula to calculate the expected value (mean) of a
random variable.
Content Limits
Items may include probability distributions or raw data. Items may include
discrete random variables, discrete uniform variables, binomial or
exponential data.
Stimulus Attribute
Items must be set in mathematical or real-world context.
Response Attributes
Items may be in fraction or decimal form.
This benchmark will be assessed using GR, FR or MC items.
Sample Item 1
1. A box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one $100 bill. A person is charged
$20 to select one bill.
a) Calculate the expected value and interpret this value in the context of this problem.
b) Is the game fair?
48
Solution:
a) $1(10/20) + $2(5/20) + $5(3/20) +$10(1/20) +$100(1/20) = $7.25 is the expected amount a person
would collect.
b) No the game is not fair because the expected winnings are less than the amount charged to play.
49
Benchmark MACC.912.S-MD.2.7
Reporting Category
Statistics and Probability
Standard
Using Probability to Make Decisions
Benchmark
Analyze decisions and strategies using probability concepts (e.g., product
testing, medical testing, pulling a hockey goalie at the end of a game).
Also assesses or
assessed by
Item Types
Benchmark
Clarification
MACC.912.S-ID.1.4, MACC.K12.MP.1.1
Content Limits
Items may include finding z-scores, confidence intervals, or p-values from
hypothesis test.
Stimulus Attribute
Items must be set in mathematical or real-world context.
Response Attributes
Items may be in fraction or decimal form.
Sample Item
Sample Item 1
This benchmark will be assessed using FR, GR, or MC items
Students will analyze decisions and strategies using the probability formulas,
and/or z-scores to solve real world statistics problems.
GR
Engineers must consider the breadths of male heads when designing
motorcycle helmets. Men have head breadths that are normally distributed
with a mean of 6.0 inches and a standard deviation of 1.0 inch (survey data
from Gordon Churchill). 100 men are randomly selected. What is the
probability that the men have a mean head breadth less than 6.2?
Sample Response
0
.
9
7
7
2