1
Benchmark MA.912.P.1.1
Reporting
Category
Probability and Statistics
Standard
Understand the counting principle, permutations, and combinations, and use
them to solve problems.
Benchmark
Number
Benchmark
MA.912.P.1.1
Also Assesses
or Assessed by
Item Types
Content Limits
Use counting principles, including the addition and the multiplication
principles, to determine size of finite sample spaces and probabilities of events
in those spaces.
N/A
This benchmark will be assessed using MC and FR items.
Items may include multiple events or repeats of one event.
Benchmark
Clarification
Students will determine probabilities of single or multiple events using
addition and multiplication counting principles.
Stimulus
Attributes
Response
Attributes
Items may be set in either real world or mathematical contexts. Items may be a
single event, multiple events, or a combination.
Responses will be in fraction 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)
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?
NEEDS ANSWER‼!
Sample Item
2
Benchmark MA.912.P.1.2
Reporting
Category
Probability and Statistics
Standard
Understand the counting principle, permutations, and combinations, and use
them to solve problems.
Benchmark
Number
Benchmark
MA.912.P.1.2
Also Assesses
or Assessed
by
Item Types
Content
Limits
Use formulas for permutations and combinations to count outcomes and
determine probabilities of events.
N/A
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.
Benchmark
Clarification
Students will determine probable outcomes of events using the permutation and
combination formulas.
Stimulus
Attributes
Response
Attributes
Sample Item
Test items may include illustrations of the following: tables, lists, tree diagrams,
and other forms.
MC items of permutations should include answers using combination formula.
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?
NEEDS ANSWER‼!
3
Benchmark MA.912.P.2.1
Reporting Category
Standard
Benchmark
Item Types
Benchmark
Clarification
Content Limits
Stimulus Attribute
Response Attributes
Probability and Statistics
Develop rules for finding probabilities of combined and
complementary events. Understand and use conditional probability
and the related Bayes’ Theorem.
MA.912.P.2.1
Determine probabilities of complementary events and calculate
odds for and against the occurrence of events.
This benchmark will be assessed using MC and FR items.
Given complementary events and frequency data, students will be
able to identify the two possible outcomes and determine their
respective probabilities.
Given an event and frequency data, students will be able to identify
the favorable and unfavorable outcomes and calculate the odds of
the event.
Less complex:
Given the odds for or against an event, students will be able to
determine the probability of the event.
Given the probability of an event, students will be able to calculate
the odds for and against the event.
Algebra 1 and Geometry EOC General Content Limits from p. 31 of
http://fcat.fldoe.org/eoc/pdf/FL10_Alg_EOC_ISB_WT_r4g.pdf
Tables and charts may be used.
Items may be set in either real-world or mathematical contexts.
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.
4
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
11
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
5
Benchmark MA.912.P.2.2
Reporting Category
Standard
Benchmark
Item Types
Benchmark
Clarification
Content Limits
Stimulus Attribute
Response Attributes
Probability and Statistics
Develop rules for finding probabilities of combined and
complementary events. Understand and use conditional probability
and the related Bayes’ Theorem.
MA.912.P.2.2
Determine probabilities of independent events.
This benchmark will be assessed using MC and FR items.
Given the frequency data for two or more independent events,
students will be able to determine the probability of a combined
event.
Given the probability of a combined event and the frequency data
for all but one of the independent events, the student will be able to
determine the probability of the remaining independent event.
Less complex:
Given the probabilities for two or more independent events, students
will be able to determine the probability of a combined event.
Given the probability of a combined event and all but one of the
probabilities of the independent events, the student will be able to
determine the probability of the remaining independent event.
Algebra 1 and Geometry EOC General Content Limits from p. 31 of
http://fcat.fldoe.org/eoc/pdf/FL10_Alg_EOC_ISB_WT_r4g.pdf
No more than 4 independent events.
Tables and charts may be used.
Items may be set in either real-world or mathematical contexts.
Gridded response items may have fractional answers.
Fractional answers must be in simplified form.
6
SAMPLE ITEM
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
7
Benchmark MA.912.P.2.3
Reporting Category
Standard
Benchmark
Item Types
Benchmark
Clarification
Probability and Statistics
Develop rules for finding probabilities of combined and
complementary events. Understand and use conditional probability
and the related Bayes’ Theorem.
MA.912.P.2.3
Understand and use the concept of conditional probability,
including: understanding how conditioning affects the
probability of events; finding conditional probabilities from a
two-way frequency table.
This benchmark will be assessed using MC and FR items.
Given a set of probabilities (e.g., complementary, combined,
conditional, independent), student will be able to calculate
conditional probabilities by creating a frequency table or applying
Bayes Theorem.
Less complex:
Given a two-way frequency table, student will be able to calculate
conditional probabilities.
Content Limits
Algebra 1 and Geometry EOC General Content Limits from p. 31 of
http://fcat.fldoe.org/eoc/pdf/FL10_Alg_EOC_ISB_WT_r4g.pdf
Stimulus Attribute
No more than 3 mutually exclusive events (e.g., A1, A2, A3) and
one independent event (e.g., B) in Bayes Theorem questions.
Tables and charts may be used.
Response Attributes
Items may be set in either real-world or mathematical contexts.
Probabilities may be given in fraction or decimal form.
Fractional answers must be in simplified form.
8
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:
Hits Florida
Misses
Florida
Defining
Rating of 5
0.005
Rating of 14
0.315
0.32
0.01
0.015
0.67
0.985
0.68
1
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
9
Benchmark MA.912.P.3.1
Body of
Knowledge
Probability and Statistics
Standard
Standard 3
Investigate probability distributions, and calculate and interpret their means an
d variances. Use and apply the normal distribution, including using the central
limit theorem.
Benchmark
MA912.P.3.1 Determine probabilities of events from distributions.
Also assesses or MA912.P.3.2 and MA912.P.3.3
assessed by
This benchmark will be assessed using FR or MC items.
Item Types
Benchmark
Clarification
Students will create and analyze probability distributions.
Content Limits
Items may include lists of items with or without their corresponding
probabilities and ask for individual probabilities or a probability distribution
to be created from the data. Items may also show a probability distribution
and ask whether it is a legitimate distribution or ask for specific information
from the distribution.
Stimulus
Attribute
Items must be set in mathematical or real-world context.
Response
Attributes
Items may be in fraction or decimal form. Items may also be a probability
distribution.
Sample Item 1
FR
A box contains five $1 bills, four $5 bills, three $10 bills, and one $20 bill. What is the probability
the $20 bill will be drawn at random when one bill is drawn?
Sample Response
1
/
1
Item Context
3
Probability and Statistics
10
Benchmark MA.912.P.3.2
Body of Knowledge
Probability and Statistics
Standard
Standard 3
Investigate probability distributions, and calculate and interpret their
means and variances. Use and apply the normal distribution, including
using the central limit theorem.
Benchmark
MA912.P.3.2
Determine the mean and variance of distributions, including: discrete
uniform (all outcomes in a finite set equally likely), binomial, normal, or
exponential.
Also assesses or
assessed by
Item Types
Benchmark
Clarification
MA912.P3.1 and MA912P.3.3
Content Limits
Stimulus Attribute
Response
Attributes
Sample Item 1
This benchmark will be assessed using FR or MC items
Students will determine the mean and variance of discrete random variables,
discrete uniform variables, and variables from binomial, normal or
exponential distributions. Students will apply appropriate formulas for
calculations. Students will understand expected value of a discrete random
variable.
Items may include probability distributions or raw data. Items may include
discrete random variables, discrete uniform variables, binomial or
exponential data. Items may include calculation of mean, expected value, or
variance.
Items must be set in mathematical or real-world context.
Items may be in fraction or decimal form.
FR
A die is rolled 240 times. What is the mean number of 2s that will be rolled?
Sample Response
4
0
Item Context
Probability and Statistics
11
Sample Item 2
MC
X
P(X)
4
0.4
5
0.3
What is the variance of the Probability Distribution?
*
A.
B.
C.
D.
Item Context
0.35
1.71
2.94
5.4
Probability and Statistics
6
0.1
8
0.15
10
0.05
12
Benchmark MA.912P.3.3
Probability and Statistics
Body of
Knowledge
Standard
Standard 3
Investigate probability distributions, and calculate and interpret their mea
ns and variances. Use and apply the normal distribution, including using t
he central limit theorem
MA912P.3.3 Apply the properties of the normal distribution.
Benchmark
Also assesses or
MA912P.3.1, MA912.P.3.2, MA912.S.3.3, MA912.S.3.4, and
assessed by
MA912S.3.6
This benchmark will be assessed using FR or MC items
Item Types
Students will identify the properties of a normal distribution, find area
Benchmark
under the standard normal distribution, find probabilities of a variable
Clarification
under a normal distribution, and find specific data values for given
percentages using the standard normal distribution.
Items may include finding area under a curve between z-scores, finding
Content Limits
z-scores with given probability or finding specific data values with given
probability. Items may include students analyzing graphs with z-sores,
data values or probabilities.
Stimulus Attribute Items must be set in mathematical or real-world context.
Items may be in decimal or percentage form. Items should specify
Response
whether the answer expected is in decimal or percentage form. Z-scores
Attributes
may be negative.
Sample Item 1
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
.
Item Context
9
5
2
1
Probability & Statistics
13
Benchmark MA.912.P.3.4
Body of
Knowledge
Standard
Probability and Statistics
Standard 3
Investigate probability distributions, and calculate and interpret their
means and variances. Use and apply the normal distribution, including
using the central limit theorem
MA912.P.3.4
Apply the Central Limit Theorem to determine the probability that a sa
mple mean will be in a certain interval.
This benchmark will be assessed using FR or MC items
Benchmark
Item Types
Also assesses
or assessed by
Benchmark
Clarification
Students will determine the probability of the sample mean being
within the interval of data given. Students will need to understand zscores and probabilities. Students need to know the difference
between the z-score formula when looking at an individual data value
and the application of the Central Limit Theorem when their sample
size is 30 or more.
Items may include raw data, intervals of data, application for
individual data from a normally distributed example or from samples
with 30 or more.
Items must be set in mathematical or real-world context.
Content Limits
Stimulus
Attribute
Response
Attributes
Items may include decimals.
Sample Item 1
FR
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
.
Item Context
9
7
7
2
Probability & Statistics
14
Benchmark MA.912.S.1.1
Reporting
Category
Probability and Statistics
Standard
Learn to define appropriate questions for research and to pose questions in a
form that can be answered by collecting and analyzing data.
MA.912.S.1.1
Benchmark
Number
Benchmark
Also Assesses
Item Types
Content
Limits
Formulate an appropriate research question to be answered by collecting data
or performing an experiment.
MA.912.S.2.1, MA.912.S.2.2, MA.912.S.2.3
This benchmark will be assessed using MC items.
Items may include research questions from surveys, observational studies, or
experiments.
Benchmark
Clarification
Students will understand the basics of questioning to collect data.
The students' consideration for the statistical questioning will include
knowledge of the sampling audience, what type of sampling to be used, and
the size of the audience as well as recognizing any bias.
Stimulus
Attributes
Response
Attributes
Sample Item
Items must be set in real-world context.
Answers need to be situations that are possible.
Which of these would NOT be an appropriate research question?
A. Do people who excerise have a lower BMI (body mass index)?
*B. Will you vote for Senator XYZ even though he is Pro-Abortion?
C. What is the age comparison of cars owned by students to cars owned by
faculty at your school?
D. Is there a difference in taste between a brand name and generic brand of
chocolate chip cookies?
15
Benchmark MA.912.S.1.2
Reporting
Category
Probability and Stastistics
Standard
Learn to define appropriate questions for research and to pose questions in a
form that can be answered by collecting and analyzing data.
MA.912.S.1.2
Benchmark
Number
Benchmark
Also Assesses
or Assessed
by
Item Types
Content
Limits
Determine appropriate and consistent standards of measurement for the data
to be collected in a survey or experiment.
N/A
This benchmark will be assessed using MC items.
Items may include qualitative or quantitative measures, discrete or continuous
variables and nominal, ordinal, interval, or ratio measures.
Benchmark
Clarification
Student will use qualitative and quantitative measures to collect data.
When using quantitative measures, students will differentiate between
discrete and continuous variables.
Students will also differentiate between nominal, ordinal, interval, and ratio
measures.
Stimulus
Attributes
Response
Attributes
Sample Item
Items must be set in real-world contexts.
Answers need to be situations that are possible.
Which of these situations uses an ordinal measure?
A. Weight of fish in Lake Eustis
B. Number of off-road vehicles sold in Lake County, Florida
*C. Ranking of weight lifters
D. Ages of students enrolled in LSCC
16
Benchmark MA.912.S.2.1
Reporting
Category
Probability and Statistics
Standard
Benchmark
Number
Benchmark
Learn key methods for collecting data and basic sampling principles.
MA.912.S.2.1
Also
Assesses or
Assessed by
Item Types
Content
Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Sample
Item
Compare the difference between surveys, experiments, and observational
studies, and what types of questions can and cannot be answered by a particular
design.
Assessed by MA.912.S.1.1
This benchmark will be assessed using MC items.
Items may include differentiating between what types of questions can and
cannot be answered by surveys, experiments, or observational studies.
Students will use and compare surveys, experiments, and observational studies
and decide which questions each is designed to answer.
Items must be set in real-world context.
Answers need to situation are possible.
"After a class receives the same lesson from the same teacher the class in
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
17
Benchmark MA.912.S.2.2
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Content Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Sample Item
Probability and Statistics
Learn key methods for collecting data and basic sampling principles.
MA.912.S.2.2
Apply the definition of random sample and basic types of sampling,
including representative samples, stratified samples, censuses.
Also assessed in MA.912.S.1.1
This benchmark will be assessed using MC items.
Items may include sampling methods such as simple random, systematic,
stratified, or cluster.
Students will choose the type of sampling from a variety of situations as well
as understand the reasoning behind random sampling.
Items must be set in real-world contexts.
Answers need to be situations that are possible.
"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. sytematic sampling
*D. stratified sampling
18
Benchmark MA.912.S.2.3
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also
Assesses or
Assessed by
Item Types
Content
Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Sample Item
Probability and Statistics
Learn key methods for collecting data and basic sampling principles.
MA.912.S.2.3
Identify sources of bias, including sampling and non-sampling errors.
Also assessed in MA.912.S.1.1
This benchmark will be assessed using MC items.
Items may include bias from both sampling and non-sampling, and intentional
or non-intentional errors.
Students will identify types of bias whether from the sample, types of questions
asked, language used, misleading or intentional ordering of questions.
Items must be set in real-world context.
Answers need to be situations that are possible.
"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
19
Benchmark MA.912.S.3.2
Reporting
Category
Standard
Benchmark
Number
Benchmark
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and strength of relationship between variables. Learn to distinguish
between different types and to select the appropriate visual form to present
different types of data.
MA.912.S.3.2
Collect, organize, and analyze data sets, determine the best format for the data and
present visual summaries.
MA.912.S.3.1
Also
Assesses or
Assessed by
Item Types This benchmark will be assessed using FR or MC items.
Content
Items for collection and organization of data may be represented in frequency
Limits
distributions with classes, frequencies, class limits, class boundaries, class width,
and class midpoint. Items for visual representations may include histogram,
frequency polygon, ogive, relative frequency graph, bar graph, pareto graph, timeseries graph, pie graph, stem and leaf plot, grouped and ungrouped frequency
distributions, or categorical frequency distributions.
Benchmark Students will collect, organize and analyze data and determine how to represent
Clarificatio the information in the correct visual summary. Students organize data with
n
frequency distributions. Students will choose and present the data in the correct
visual representation.
Stimulus
Items must be set in real-world context.
Attributes
Response
Answers will be either in visual representations for students to identify or they
Attributes
will analyze data given in a visual representation.
20
Sample
Item
Sample Item 1
MC
In a study of a sample of 200 student completion times for a state test the
following times were recorded (in minutes).
29.5-39.5
5
39.5-49.5
24
49.5-59.5
35
59.5-69.5
72
69.5-79.5
36
79.5-89.5
28
What are the class boundaries?
A.
*B.
C.
D.
29-39, 39-49, 49-59, 59-69, 69-79, 79-89
29.5-39.5, 39.5-49.5, 49.5-59.5, 59.5-69.5, 69.5-79.5
30-40, 40-50, 50-60, 60-70, 70-80, 80-90
34.5, 44.5, 54.5, 64.5, 74.5, 84.5
Sample Response 1 B
Item Context:
Mathematics (Statistics)
Sample Item 2
MC
In a study of a sample of 200 student completion times for a state test the
following times were recorded (in minutes).
29.5-39.5
5
39.5-49.5
24
49.5-59.5
35
59.5-69.5
72
69.5-79.5
36
79.5-89.5
28
What are the class limits?
A.
B.
*C.
D.
29-39, 39-49, 49-59, 59-69, 69-79, 79-89
29.5-39.5, 39.5-49.5, 49.5-59.5, 59.5-69.5, 69.5-79.5
30-40, 40-50, 50-60, 60-70, 70-80, 80-90
34.5, 44.5, 54.5, 64.5, 74.5, 84.5
Sample Response 2 C
Item Context:
Mathematics (Statistics)
21
Benchmark MA.912.S.3.3
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Content Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and stength of relationship between variables. Learn to
distinguish between different types and to select the appropriate visual form to
present different types of data.
MA.912.S.3.3
Calculate and interpret measures of the center of a set of data, including mean,
median, and weighted mean, and use these measures to make comparisons
among sets of data.
MA912.P.3.3
This benchmark will be assessed using FR or MC items.
Items may include calculations for mean, median, mode, or weighted mean of
a data set. Items may include the choice of which measure of center is the best
representation of the data set.
Students will calculate, summarize, and interpret data using measures of center
including mean, median, mode, and weigthed mean. Students will use these
measures to make comparisons and draw conclusions about data sets.
Items must be set in mathematical or real-world context.
Answers for the mean will follow the Rounding Rule for the Mean (rounded
to one more decimal place than the raw data).
22
Sample Item
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
*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)
23
Benchmark MA.912.S.3.4
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Content Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and stength of relationship between variables. Learn to
distinguish between different types and to select the appropriate visual form to
present different types of data.
MA.912.S.3.4
Calculate and interpret measures of variance and standard deviation. Use these
measures to make comparisons among sets of data.
MA912.P.3.3
This benchmark will be assessed using FR or MC items.
Items may include calculating variance and/or standard devaition for grouped
or ungrouped data. Items may also include uses of variance and standard
deviation such as determining spread of data, comparing two or more data sets,
determining consistency of a variable, or determining number of data beween
specified intervals.
Students will use their calculations of the measures of variance and standard
deviation to interpret, describe and compare sets of data.
Items must be set in mathematical or real-world context.
Answers for the standard deviation will follow the Rounding Rule for the
Standard Deviation(rounded to one more decimal place than the raw data).
24
Sample Item
Sample Item 1 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 1
Item Context
1
9
.
2
719.2
Mathematics (Statistics)
Sample Item 2 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
Sample Response 2
Item Context
1
7
2
4
9
517249
Mathematics (Statistics)
25
Benchmark MA.912.S.3.5
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Content Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Sample Item
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and strength of relationship between variables. Learn to
distinguish between different types and to select the appropriate visual form to
present different types of data.
MA.912.S.3.5
Calculate and interpret the range and quartiles of a set of data.
N/A
This benchmark will assessed using FR or MC items.
Items may include calculations of range, percentile, quartile and decile,
calculation of the inter-quartile range, and the interpretation of the data
according to the data set. Items may include box-and-whisker plots.
Students will identify the position of a data value in a data set using measures
of position such as range, standard scores, percentiles, quartiles and deciles.
Students will be able to apply the five number summary in a box-and-whisker
plot.
Items must be set in mathematical or real-world context.
Answers will use rules of procedures for percentile calculations. Answers in
box-and-whisker plots will have visible numbers for the 5 number summary.
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)
26
Benchmark MA.912.S.3.6
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also
Assesses or
Assessed by
Item Types
Content
Limits
Benchmark
Clarificatio
n
Stimulus
Attributes
Response
Attributes
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and strength of relationship between variables. Learn to distinguish
between different types and to select the appropriate visual form to present
different types of data.
MA.912.S.3.6
Use empirical rules (e.g. 68-95-99.7 rule) to estimate spread of distributions and
to make comparisons among sets of data.
MA912.P.3.3
This benchmark will be assessed using FR or MC items.
Items may include finding area under curve for specific data where z-scores need
to be calculated or calculating a z-score and then finding data values with specific
probability from a data set . Items may also include checking data for normality.
Students will apply the empirical rule of the Theoretical Normal Distribution to
make comparisons of data sets.
Items must be set in mathematical or real-world context.
Answers will be represented in z-scores, area (percentages) under the curve, data
values represented by specific z-scores or probability, or data normality
determination.
27
Sample
Item
Sample Item 1
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 1
A.
-1.99
*B.
1.99
C.
6.36
D.
80.9
Sample Response 1 B
Item Context:
Mathematics (Statistics)
Sample Item 2
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 2
2
.
5
6
Sample Response 2.56
Item Context:
Mathematics (Statistics)
28
Benchmark MA.912.S.3.7
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also
Assesses or
Assessed
by
Item Types
Content
Limits
Benchmark
Clarificatio
n
Stimulus
Attributes
Response
Attributes
Sample
Item
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and strength of relationship between variables. Learn to distinguish
between different types and to select the appropriate visual form to present
different types of data.
MA.912.S.3.7
Calculate the correlation coefficient of a set of paired data, and interpret the
coefficient as a measure of the strength and direction of the relationship between
the variables.
N/A
This benchmark will be assessed using FR or MC items.
Items may include calculating the correlation coefficient of a set of paired data and
the interpretation of the relationship between the two variables.
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 scatterplots to aid their interpretation of the data.
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
258 264 402 253 111 648 288 654 256 60 590
Population 4
6
9
6
3
29 15 38 20 6 81
Sample Response 1
0
Sample Response 1
Item Context:
.
7
2
7
0.727
Mathematics (Statistics)
29
Benchmark MA.912.S.3.8
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also Assesses
Item Types
Content Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Sample Item
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and strength of relationship between variables. Learn to
distinguish between different types and to select the appropriate visual form to
present different types of data.
MA.912.S.3.8
Determine whether a data distribution is symmetric or skewed based on an
appropriate graphical presentation of the data.
N/A
This benchmark will be assessed using MC items.
Items may include determining whether data is symmetric or skewed from
graphical representations in histograms or box-and-whiskerplots. Items may
also include interpretation of left or right skewed data and the reasons for
skewness determination.
Students will determine whether data distributions in histograms or box-andwhisker plots are symmetric or skewed and the reasons for skewness.
Items must be set in real-world context.
Answers need to be situations that are possible.
Sample Item 1
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)
30
Benchmark MA.912.S.3.9
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Content Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Sample Item
Probability and Statistics
Learn to work with summary measures of sets of data, including measures of
center, spread, and strength of relationship between variables. Learn to
distinguish between different types and to select the appropriate visual form to
present different types of data.
MA.912.S.3.9
Identify outliers in a set of data based on an appropriate graphical presentation
of the data, and describe the effect of outliers on the mean, median, and range
of the data.
N/A
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 will an
added higher or lower value have on a particular mean, median, mode or
range.
Students will identify outliers, the possible reasons for them and what effect
they have on the mean, median, mode and range of data.
Items must be set in mathematical or real-world context.
Answers will include numerical outliers, reasons for outliers, or effect of
specific outlier on mean, median, mode or range of data.
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
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.
Sample Response 1: B
Item Context: Mathematics (Statistics)
31
Benchmark MA.912.S.4.1
Reporting
Category
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Content Limits
Benchmark
Clarification
Stimulus
Attributes
Response
Attributes
Sample Item
Probability and Statistics
Learn to use simulations of standard sampling distributions to determine
confidence levels and margins of error. Develop measures of association
between two numerical or categorical variables. Use technological tools to
find equations of regression lines and correlation coefficients.
MA.912.S.4.1
Explain and interpret the concepts of confidence level and "margin of error".
MA 912 S 4.4 and MA912.S.5.1
This benchmark will be assessed using FR and MC items.
Items may include point estimate (best point estimate in the sample mean)
which should be unbiased, consistent, and relatively efficient. 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.
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").
Items must be set in real-world context.
Answers will include confidence levels in percentages, confidence intervals
with a mean or proportion estimate or sample sizes in whole numbers.
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
Item Context Mathematics (Statistics)
32
Benchmark MA.912.S.4.3
Body of Knowledge
Standard
Benchmark
Also Assesses
Item Types
Benchmark
Clarification
Content Limits
Stimulus Attribute
Response Attributes
Sample Item 1
Sample Response
Item Context
Statistics
Standard 4
Apply the Central Limit Theorem to solve problems.
MA.912.P.3.4, MA.912.S.4.1 , MA.912.S.4.2
33
Benchmark MA.912.S.4.4
Body of Knowledge
Standard
Benchmark
Also Assesses
Item Types
Benchmark
Clarification
Content Limits
Stimulus Attribute
Response Attributes
Sample Item 1
Sample Response
Item Context
Statistics
Standard 4
Approximate confidence intervals for means using simulations of the
distribution of the sample mean.
MA.912.S.4.1
This benchmark will be assessed using FR and MC items.
34
Benchmark MA.912.S.4.5
Reporting Category
Standard
Benchmark
Item Types
Benchmark
Clarification
Content Limits
Probability and Statistics
Learn to use simulations of standard sampling distributions to
determine confidence levels and margins of error. Develop
measures of association between two numerical or categorical
variables. Use technological tools to find equations of regression
lines and correlation coefficients.
MA.912.S.4.5
Find the equation of the least squares regression line for a set of
data.
This benchmark will be assessed using MC and FR items.
(None needed)
Algebra 1 and Geometry EOC General Content Limits from p. 31 of
http://fcat.fldoe.org/eoc/pdf/FL10_Alg_EOC_ISB_WT_r4g.pdf
5 <= n <= 8 if student is calculating sums by hand (i.e., without the
use of Excel or other software)
Stimulus Attributes
Data points limited to one, two, or three digits
Tables and charts may be used.
Response Attributes
Items may be set in either real-world or mathematical contexts.
Equations may be presented in standard or slope-intercept form.
35
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, what is the equation for the least squares regression line?
Classroom Test
Score
90
80
75
40
80
85
A)
B)
C)
D)
Algebra EOC Scale
Score
450
425
425
350
430
440
y = .11x + 14.73
y = 14.73x + .11
y = 271.17x + 1.98
y = 1.98x + 271.17
Answer:
D)
Solution:
(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
36
Benchmark MA.912.S.5.1
Body of
Knowledge
Standard
Benchmark
Also assesses
or assessed by
Item Types
Benchmark
Clarification
Content Limits
Stimulus
Attribute
Response
Attributes
Probability and Statistics
Standard 5
Gather data and determine confidence intervals to make inferences
about means, and use hypothesis tests to make decisions. Learn to use
data to approximate p-values and to determine whether correlations
between variables are significant.
MA912.S.5.1
Analyze the relationship between confidence level, margin of error and
sample size.
MA 912 S 4.1 and MA 912 S 4.4
This benchmark will be assessed using FR and MC items.
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 als
o interpret and apply the maximum error of the estimate (ʺmargin of
errorʺ). Students will need to understand what steps to follow to
compensate for the confidence interval, margin of error or sample size
when one or two of the three factors changes.
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.
Items must be set in mathematical or real-world context.
Items may include decimals for point estimate and percentages for
confidence level. Sample size must be a whole number.
37
Sample Item 1
FR
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
Item Context
0
4
Probability & Statistics
38
Benchmark MA.912.S.5.2
Body of
Knowledge
Standard
Probability and Statistics
Benchmark
MA912.S.5.2
Apply the general principles of hypothesis testing.
Also assesses or
assessed by
MA 912 S 5.3 and MA 912 5.4
Item Types
This benchmark will be assessed using and MC items.
Benchmark
Clarification
Content Limits
Students will understand, create and analyze the null hypothesis and the
alternative hypothesis (basics of hypothesis testing).
Items may include writing a null and alternative hypothesis for data or
interpreting a given null and alternative hypothesis.
Items must be set in mathematical or real-world context.
Stimulus
Attribute
Response
Attributes
Sample Item 1
Standard 5
Gather data and determine confidence intervals to make inferences about
means, and use hypothesis tests to make decisions. Learn to use data to
approximate p-values and to determine whether correlations between
variables are significant. .
Items may be writing the null and alternative hypothesis or analyzing the
given null and alternative hypothesis.
MC
The average age for community college students is 24.6 years.
What is the null and alternative hypothesis for the conjecture?

*
A.
B.
C.
D.
Item Context
H˳: µ= 24.6
H˳: µ < 24.6
H˳: µ =24.6
H˳:µ = 24.6
and
and
and
and
Ң: µ > 24.6
Ң: µ = 24.6
Ң: µ < 24.6
Ң: µ ≠ 24.6
Probability & Statistics
39
Benchmark MA.912.S.5.3
Body of
Knowledge
Standard
Benchmark
Also assesses or
assessed by
Item Types
Benchmark
Clarification
Content Limits
Stimulus
Attribute
Response
Attributes
Probability and Statistics
Standard 5
Gather data and determine confidence intervals to make inferences
about means, and use hypothesis tests to make decisions. Learn to
use data to approximate p-values and to determine whether
correlations between variables are significant.
MA912.S.5.3
Explain and identify the following: null hypothesis, alternative hypotheses, Type I error, and Type II error.
MA 912 S 5.2 and MA 912 S 5.4
This benchmark will be assessed using and MC items
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.
Items may include information from a hypothesis test and/or a
comparison of the two types of errors which may occur.
Items must be set in mathematical or real-world context.
Items may include a choice between the Type I and Type II error or
may include the two types of error which can occur.
40
Sample Item 1
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
41
Benchmark MA.912.S.5.4
Body of Knowledge Probability and Statistics
Standard
Standard 5
Gather data and determine confidence intervals to make inferences
about means, and use hypothesis tests to make decisions. Learn to
use data to approximate p-values and to determine whether
correlations between variables are significant.
Benchmark
MA912.S.5.4
Explain the meaning of p-value and its role in hypothesis testing.
Also assesses or
MA912.S.5.2 and MA912.S.5.3
assessed by
Item Types
This benchmark will be assessed using FR and MC items.
Benchmark
Students will understand the p-value is the probability of getting a
Clarification
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.
Content Limits
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 Items must be set in mathematical or real-world context.
Response
Answers may include the p-value and/or a decision to be made
Attributes
based on the p-value.
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. At a 0.05 level of significance, what is the p-value of this sample?
Sample Response
0.
0
Item Context
2
7
4
Probability & Statistics
42
Sample Item 2
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