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Mathematics: Applications and Interpretations II
5 Credits
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Grade 12: Unit Three Statistics
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IB Learner ProfileIB Programs aim to develop internationally minded people who are striving to become:
Inquirers Their natural curiosity is nurtured. They acquire the skills necessary to conduct constructive inquiry and research, and become independent active learners. They actively enjoy learning and this love of learning will be sustained throughout their lives.
Knowledgeable They explore concepts, ideas and issues, which have global relevance and importance. In so doing, they acquire, and are able to make use of, a significant body of knowledge across a range of disciplines.
Critical thinkers They exercise initiative in applying thinking skills critically and creatively to approach complex problems and make reasoned decisions.
Communicators They understand and express ideas and information confidently and creatively in more than one language and in a variety of modes of communication.
Risk-takers They approach unfamiliar situations with confidence and forethought, and have the independence of spirit to explore new roles, ideas and strategies. They are courageous and articulate in defending those things in which they believe.
Principled They have a sound grasp of the principles of moral reasoning. They have integrity, honesty, a sense of fairness and justice and respect for the dignity of the individual.
Caring They show empathy, compassion and respect towards the needs and feelings of others. They have a personal commitment to action and service to make a positive difference to the environment and to the lives of others.
Open-minded Through an understanding and appreciation of their own culture, they are open to the perspectives, values and traditions of other individuals and cultures and are accustomed to seeking and considering a range of points of view.
Well-balanced They understand the importance of physical and mental balance and personal wellbeing for themselves and others. They demonstrate perseverance and self-discipline.
Reflective They consider their own learning and personal development. They are able to analyze their strengths and weaknesses in a constructive manner.
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Course DescriptionThis course recognizes the increasing role that mathematics and technology play in a diverse range of fields in a data-rich world. As such, it emphasizes the meaning of mathematics in context by focusing on topics that are often used as applications or in mathematical modelling. To give this understanding a firm base, this course also includes topics that are traditionally part of a pre-university mathematics course such as calculus and statistics.
The course makes extensive use of technology to allow students to explore and construct mathematical models. Mathematics: applications and interpretation will develop mathematical thinking, often in the context of a practical problem and using technology to justify conjectures. Students should enjoy seeing mathematics used in real-world contexts and to solve real-world problems.
International Baccalaureate Organization. (2019). Mathematics: applications and interpretation guide (First assessment 2021) [Curriculum Guide]. Wales, United Kingdom: International Baccalaureate Organization (UK) Ltd.
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# Topic Suggested Timing
1 Introduction to Trigonometry 6 weeks
2 Probability 10 weeks
3 Statistics 10 weeks
4 Introduction to Calculus 10 weeks
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Educational TechnologyStandards
8.1.8.A.1, 8.1.8.B.1, 8.1.8.C.1, 8.1.8.D.1, 8.1.8.E.1, 8.1.8.F.1
Technology Operations and Concepts Create professional documents (e.g., newsletter, personalized learning plan, business letter or flyer) using advanced features of a
word processing program.
Creativity and Innovation Synthesize and publish information about a local or global issue or event on a collaborative, web-based service.
Communication and Collaboration Participate in an online learning community with learners from other countries to understand their perspectives on a global problem
or issue, and propose possible solutions.
Digital Citizenship Model appropriate online behaviors related to cyber safety, cyber bullying, cyber security, and cyber ethics.
Research and Information Literacy Gather and analyze findings using data collection technology to produce a possible solution for a content-related or real-world
problem.
Critical Thinking, Problem Solving, Decision Making Use an electronic authoring tool in collaboration with learners from other countries to evaluate and summarize the perspectives of
other cultures about a current event or contemporary figure.
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21st Century Life & Career SkillsStandards:
9.1.8.A.1, 9.1.8.A.2, 9.1.8.B.1, 9.1.8.C.1, 9.1.8.C.2, 9.1.8.C.3, 9.1.8.D.2, 9.1.8.D.3, 9.3.8.B.3
Learning and Innovation Skills: Creativity and Innovation
Use multiple points of view to create alternative solutions.
Critical Thinking and Problem Solving Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and problem-solving skills. Implement problem-solving strategies to solve a problem in school or the community.
Communication and Collaboration Skills Determine an individual’s responsibility for personal actions and contributions to group activities. Demonstrate the use of compromise, consensus, and community building strategies for carrying out different tasks, assignments,
and projects. Model leadership skills during classroom and extra-curricular activities.
Cross-Cultural Understanding and Interpersonal Communication Demonstrate the ability to understand inferences. Use effective communication skills in face-to-face and online interactions with peers and adults from home and from diverse
cultures.
Career Exploration Evaluate personal abilities, interests, and motivations and discuss how they might influence job and career selection.
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Career Ready PracticesCareer Ready Practices describe the career-ready skills that all educators in all content areas should seek to develop in their students. They are practices that have been linked to increase college, career, and life success. Career Ready Practices should be taught and reinforced in all career exploration and preparation programs with increasingly higher levels of complexity and expectation as a student advances through a program of study.
CRP1. Act as a responsible and contributing citizen and employee Career-ready individuals understand the obligations and responsibilities of being a member of a community, and they demonstrate this understanding every day through their interactions with others. They are conscientious of the impacts of their decisions on others and the environment around them. They think about the near-term and long-term consequences of their actions and seek to act in ways that contribute to the betterment of their teams, families, community and workplace. They are reliable and consistent in going beyond the minimum expectation and in participating in activities that serve the greater good.
CRP2. Apply appropriate academic and technical skills. Career-ready individuals readily access and use the knowledge and skills acquired through experience and education to be more productive. They make connections between abstract concepts with real-world applications, and they make correct insights about when it is appropriate to apply the use of an academic skill in a workplace situation.
CRP3. Attend to personal health and financial well-being. Career-ready individuals understand the relationship between personal health, workplace performance and personal well-being; they act on that understanding to regularly practice healthy diet, exercise and mental health activities. Career-ready individuals also take regular action to contribute to their personal financial well-being, understanding that personal financial security provides the peace of mind required to contribute more fully to their own career success.
CRP4. Communicate clearly and effectively and with reason. Career-ready individuals communicate thoughts, ideas, and action plans with clarity, whether using written, verbal, and/or visual methods. They communicate in the workplace with clarity and purpose to make maximum use of their own and others’ time. They are excellent writers; they master conventions, word choice, and organization, and use effective tone and presentation skills to articulate ideas. They are skilled at interacting with others; they are active listeners and speak clearly and with purpose. Career-ready individuals think about the audience for their
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Career Ready Practicescommunication and prepare accordingly to ensure the desired outcome. CRP5. Consider the environmental, social and economic impacts of decisions. Career-ready individuals understand the interrelated nature of their actions and regularly make decisions that positively impact and/or mitigate negative impact on other people, organization, and the environment. They are aware of and utilize new technologies, understandings, procedures, materials, and regulations affecting the nature of their work as it relates to the impact on the social condition, the environment and the profitability of the organization.
CRP6. Demonstrate creativity and innovation. Career-ready individuals regularly think of ideas that solve problems in new and different ways, and they contribute those ideas in a useful and productive manner to improve their organization. They can consider unconventional ideas and suggestions as solutions to issues, tasks or problems, and they discern which ideas and suggestions will add greatest value. They seek new methods, practices, and ideas from a variety of sources and seek to apply those ideas to their own workplace. They take action on their ideas and understand how to bring innovation to an organization.
CRP7. Employ valid and reliable research strategies. Career-ready individuals are discerning in accepting and using new information to make decisions, change practices or inform strategies. They use reliable research process to search for new information. They evaluate the validity of sources when considering the use and adoption of external information or practices in their workplace situation.
CRP8. Utilize critical thinking to make sense of problems and persevere in solving them. Career-ready individuals readily recognize problems in the workplace, understand the nature of the problem, and devise effective plans to solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully investigate the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a solution is agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others.
CRP9. Model integrity, ethical leadership and effective management. Career-ready individuals consistently act in ways that align personal and community-held ideals and principles while employing strategies to positively influence others in the workplace. They have a clear understanding of integrity and act on this understanding in every decision. They use a variety of means to positively impact the directions and actions of a team or organization, and they apply insights into human behavior to change
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Career Ready Practicesothers’ action, attitudes and/or beliefs. They recognize the near-term and long-term effects that management’s actions and attitudes can have on productivity, morals and organizational culture.
CRP10. Plan education and career paths aligned to personal goals. Career-ready individuals take personal ownership of their own education and career goals, and they regularly act on a plan to attain these goals. They understand their own career interests, preferences, goals, and requirements. They have perspective regarding the pathways available to them and the time, effort, experience and other requirements to pursue each, including a path of entrepreneurship. They recognize the value of each step in the education and experiential process, and they recognize that nearly all career paths require ongoing education and experience. They seek counselors, mentors, and other experts to assist in the planning and execution of career and personal goals.
CRP11. Use technology to enhance productivity. Career-ready individuals find and maximize the productive value of existing and new technology to accomplish workplace tasks and solve workplace problems. They are flexible and adaptive in acquiring new technology. They are proficient with ubiquitous technology applications. They understand the inherent risks-personal and organizational-of technology applications, and they take actions to prevent or mitigate these risks.
CRP12. Work productively in teams while using cultural global competence. Career-ready individuals positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members. They plan and facilitate effective team meetings.
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Differentiated Instruction Accommodate Based on Students Individual Needs: Strategies
Time/General
Extra time for assigned tasks
Adjust length of assignment
Timeline with due dates for reports and projects
Communication system between home and school
Provide lecture notes/outline
Processing
Extra Response time
Have students verbalize steps
Repeat, clarify or reword directions
Mini-breaks between tasks
Provide a warning for transitions
Reading partners
Comprehension
Precise step-by-step directions
Short manageable tasks
Brief and concrete directions
Provide immediate feedback
Small group instruction
Emphasize multi-sensory learning
Recall
Teacher-made checklist
Use visual graphic organizers
Reference resources to promote independence
Visual and verbal reminders
Graphic organizers
Assistive Technology
Computer/whiteboard
Tape recorder
Spell-checker
Audio-taped books
Tests/Quizzes/Grading
Extended time
Study guides
Shortened tests
Read directions aloud
Behavior/Attention
Consistent daily structured routine
Simple and clear classroom rules
Frequent feedback
Organization
Individual daily planner
Display a written agenda
Note-taking assistance
Color code materials
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EnrichmentAccommodate Based on Students individual Needs: Strategies
Adaption of Material and Requirements
Evaluate Vocabulary
Elevated Text Complexity
Additional Projects
Independent Student Options
Projects completed individual or with Partners
Self Selection of Research
Tiered/Multilevel Activities
Learning Centers
Individual Response Board
Independent Book Studies
Open-ended activities
Community/Subject expert mentorships
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AssessmentsSuggested Formative/Summative Classroom Assessments
Timelines, Maps, Charts, Graphic Organizers
Teacher-created Unit Assessments, Chapter Assessments, Quizzes
Accountable Talk, Debate, Oral Report, Role Playing, Think Pair, and Share
Projects, Portfolio, Presentations, Prezi, Gallery Walks
Homework
Concept Mapping
Primary and Secondary Source analysis
Photo, Video, Political Cartoon, Radio, Song Analysis
Create an Original Song, Film, or Poem
Glogster to make Electronic Posters
Internal and External IB Assessments
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Interdisciplinary Connections English Language Arts
Journal Writing Close reading of industry-related content Create a brochure for a specific industry Keep a running word wall of industry vocabulary
Social Studies Research the history of a given industry/profession Research prominent historical individuals in a given industry/profession Use historical references to solve problems
World Language Translate industry-content Create a translated index of industry vocabulary Generate a translated list of words and phrases related to workplace safety
Math Research industry salaries for a geographic area and juxtapose against local cost of living Go on a geometry scavenger hunt Track and track various data, such as industry’s impact on the GDP, career opportunities or among of individuals currently occupying careers
Fine & Performing Arts Create a poster recruiting young people to focus their studies on a specific career or industry Design a flag or logo to represent a given career field
Science Research the environmental impact of a given career or industry Research latest developments in industry technology Investigate applicable-careers in STEM fields
Course: Applications and Interpretations SL Topic: Statistics14 | P a g e
Unit 2 Statistics
Grade Level: 12
Description: Statistics is concerned with the collection, analysis and interpretation of quantitative data and uses the theory of probability to estimate parameters, discover empirical laws, test hypotheses and predict the occurrence of events. Statistical representations and measures allow us to represent data in many different forms to aid interpretation. The aim of the standard level content in the statistics and probability topic is to introduce students to important concepts, techniques and representations used in statistics and probability and their meaningful application in the real world. Students should be given the opportunity to approach this topic in a practical way, to understand why certain techniques are used and to interpret the results. The use of technology such as simulations, spreadsheets, statistics software and statistics apps can greatly enhance this topic.It is expected that most of the calculations required will be carried out using technology, but explanations of calculations by hand may enhance understanding. The emphasis is on choosing the most appropriate technique, and understanding and interpreting the results obtained in context.
In examinations students should be familiar with how to use the statistics functionality of allowed technology. Data sets will be considered to be the population unless otherwise stated.
Suggested concepts embedded in this topic:Quantity, validity, approximation, modelling, relationships, patterns.
New Jersey Core Curriculum Content Standards (NJCCCS): NJSLS.S.ID.A, NJSLS.S.ID.B, NJSLS.S.ID.C, NJSLS.S.IC.A, NJSLS.S.IC.B, & MP 1-8.
NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
1. Investigate unfamiliar How can you understand Concepts of Activity: Students research Texas Instruments TI84
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
situations, both abstract and from the real world, involving organizing and analyzing population parameters based on sampling techniques, making conjectures, drawing conclusions, and testing their validity. S.IC.A, S.IC.B
and evaluate random processes underlying statistical experiments?
How can you make inferences and justify conclusions from sample surveys, experiments, and observational studies
population, sample, random sample, discrete and continuous data. This is designed to cover the key questions that students should ask when they see a data set/analysis.
Reliability of data sources and bias in sampling. Dealing with missing data, errors in the recording of data.
Interpretation of outliers. Outlier is defined as a data item which is more than 1.5 × interquartile range (IQR) from the nearest quartile.Awareness that, in context, some outliers are a valid part of the sample but some outlying data items may be an error in the sample. Link to: box
news articles containing misleading statistics and debate the ethical implications of their research.
Summative and Formative Assessments (Quizzes & Tests) for each topic.
Homework and Classwork assignments based on daily lessons.
Links to other subjects: Descriptive statistics and random samples (biology, psychology, sports exercise and health science, environmental systems and societies, geography, economics; business management); research methodologies (psychology).
Aim 8: Misleading statistics; examples of problems caused by absence of representative samples, for example Google
Plus Graphing Display Calculator
IB Question bank
www.geogebra.com
www.Khanacademy.org
www.illustrativemathem atics.org
www.illuminations.nctm .org
www.youtube.com
http://ocw.mit.edu
www.NJCTL.org
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
and whisker diagrams (SL4.2) and measures of dispersion (SL4.3).
Sampling techniques and their effectiveness. Simple random, convenience, systematic, quota and stratified sampling methods.
flu predictor, US presidential elections in 1936, Literary Digest v George Gallup, Boston “pot-hole” app.
International-mindedness: The Kinsey report–famous sampling techniques.
TOK: Why have mathematics and statistics sometimes been treated as separate subjects? How easy is it to be misled by statistics? Is it ever justifiable to purposely use statistics to mislead others?
2. Transform common realistic contexts into statistical representations of data; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation; use
How can you summarize, represent, and interpret data on a single count or measurement variable?
What effects can an outlier have on a set of univariate data?
What are the different formulae used to calculate variance and
Presentation of data (discrete and continuous): frequency distributions (tables).Class intervals will be given as inequalities, without gaps.
Histograms.Cumulative frequency;
Activity: Textbook practice problems
Links to other subjects: Presentation of data (sciences, individuals and societies).
International-mindedness: Discussion of the different formulae for the same statistical measure (for
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
appropriate notation and terminology.
S.IC.A, S.IC.B
.
what are the implications on the interpretation of the variance statistic?
cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).Frequency histograms with equal class intervals.Not required: Frequency density histograms.
Production and understanding of box and whisker diagrams.Use of box and whisker diagrams to compare two distributions, using symmetry, median, interquartile range or range. Outliers should be indicated with a cross.Determining whether the data may be normally distributed by consideration of
example, variance).
TOK: What is the difference between information and data? Does “data” mean the same thing in different areas of knowledge?
Summative and Formative Assessments (Quizzes & Tests) for each topic.
Homework and Classwork assignments based on daily lessons.
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
the symmetry of the box and whiskers.
3. Recall, select and use their knowledge of measures of central tendency and dispersion, concepts and techniques to compare a variety of familiar and unfamiliar contexts.
S.ID.A and MP1-8.
How can you compare measures of central tendency and dispersion in population or sample data?
What are the effects on measures of central tendency and dispersion when a data set is changed by adding a common difference or multiplying by a common ratio?
TOK: Could mathematics make alternative, equally true, formulae? What does this tell us about mathematical truths? Does the use of statistics lead to an over-emphasis on attributes that can be easily measured over
Measures of central tendency (mean, median and mode).Estimation of mean from grouped data.Calculation of mean using formula and technology.Students should use mid-interval values to estimate the mean of grouped data.
Modal class.For equal class intervals only.
Measures of dispersion (interquartile range, standard deviation and variance).Calculation of standard deviation and variance of the sample using only technology, however hand calculations may enhance
Activity: Getting a feel for data task from IB Teacher Support Materials.
Summative and Formative Assessments (Quizzes & Tests) for each topic.
Homework and Classwork assignments based on daily lessons.
Other contexts: Comparing variation and spread in populations, human or natural, for example agricultural crop data, social indicators, reliability and maintenance.
Links to other subjects: Descriptive statistics (sciences and individuals and societies); consumer price index (economics).
International-mindedness: The benefits of sharing and analyzing data from different
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
those that cannot? understanding. Variance is the square of the standard deviation
Effect of constant changes on the original data.Examples: If three is subtracted from the data items, then the mean is decreased by three, but the standard deviation is unchanged.If all the data items are doubled, the mean is doubled and the standard deviation is also doubled.
Quartiles of discrete data.Using technology. Awareness that different methods for finding quartiles exist and therefore the values obtained using technology and by hand may differ.
countries; discussion of the different formulae for variance.
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
4. Transform bivariate data from common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation; use appropriate notation and terminology.S.ID.B & MP1-8.
TOK: Correlation and causation–can we have knowledge of cause and effect relationships given that we can only observe correlation? What factors affect the reliability and validity of mathematical models in describing real-life phenomena?
Linear correlation of bivariate data.Pearson’s product-moment correlation coefficient, r.Technology should be used to calculate r. However, hand calculations of r may enhance understanding.Critical values of r will be given where appropriate.Students should be aware that Pearson’s product moment correlation coefficient (r) is only meaningful for linear relationships.
Scatter diagrams; lines of best fit, by eye, passing through the mean point.Positive, zero, negative; strong, weak, no correlation.
Activity: Anscombe’s quartet–cognitive activator from IB Teacher Support Materials.
Predicting heights of young children. Generate the data by using the http://www.babycenter.com/child-height-predictor
Summative and Formative Assessments (Quizzes & Tests) for each topic.
Homework and Classwork assignments based on daily lessons.
Other contexts: Linear regressions where correlation exists between two variables. Exploring cause and dependence for categorical variables, for example, on what factors might political persuasion depend?
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
Students should be able to make the distinction between correlation and causation and know that correlation does not imply causation.
Equation of the regression line of y on x.Use of the equation of the regression line for prediction purposes.Interpret the meaning of the parameters, a and b, in a linear regression y=ax+b.Technology should be used to find the equation.Students should be aware of the dangers of extrapolation; that they cannot always reliably make a prediction of x from a value of y, when
Links to other subjects: Curves of best fit, correlation and causation (sciences); scatter graphs (geography).
Aim 8: The correlation between smoking and lung cancer was “discovered” using mathematics. Science had to justify the cause.
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
using a yon x line.
5. Investigate unfamiliar situations, both abstract and from the real world, involving organizing and analyzing information, making conjectures, drawing conclusions, and testing their validity using Spearman’s rank correlation coefficient, r s.
S.ID.B & MP1-8.
When is it appropriate to Spearman’s correlation coefficient?
How does Spearman’s correlation coefficient compare to Pearson’s product moment correlation coefficient?
How can you test for correlation of non-linear data?
TOK: Does correlation imply causation? Mathematics and the world. Given that a set of data may be approximately fitted by a range of curves, where would a mathematician seek for knowledge of which equation is the “true” model?
Spearman’s rank correlation coefficient, r s.
In examinations Spearman’s rank correlation coefficient, r s, should be found using technology.
If data items are equal, ranks should be averaged.
Awareness of the appropriateness and limitations of Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient, and the effect of outliers on each.
Students should be aware that Pearson’s product moment correlation coefficient is useful when testing for only linearity and Spearman’s correlation coefficient for any monotonic relationship.
Spearman’s correlation
Activity: Anscombe’s quartet–cognitive activator from IB Teacher Support Materials.
Summative and Formative Assessments (Quizzes & Tests) for each topic.
Homework and Classwork assignments based on daily lessons.
Links to other subjects: Fieldwork (biology, psychology, environmental systems and societies, sports exercise and health science)
Aim 8: The physicist Frank Oppenheimer wrote: “Prediction is dependent only on the assumption that observed patterns will be repeated”. This is the danger of extrapolation. There are
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
coefficient is less sensitive to outliers than Pearson’s product moment correlation coefficient.
Not required: Derivation/proof of Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient.
many examples of its failure in the past, for example share prices, the spread of disease, climate change.
Links to websites: www.wikihow.com/Calculate-Spearman%27s-Rank-Correlation-Coefficient
External website: Use of databases such as Gapminder.
6. Utilize the Chi-square test for independence; formulate the null and alternative hypotheses, significance levels, contingency tables, expected frequencies, degrees of freedom, and p-values t-tests.
S.ID.B & MP1-8.
How can you formulate a null and alternative hypothesis?
What are the different experiment designs and when is it appropriate to use them?
How can significance levels and p-values help you analyze the results on an experiment?
How can you compare a p -value to the given significance level?
Formulation of null and alternative hypotheses, H0 and H1.
Significance levels.
p -values.
Students should express H0 and H1 as an equation or inequality, or in words as appropriate.
Expected and observed frequencies.
The χ2 test for independence: contingency tables, degrees of freedom, critical value.
Activity: Chi square test for independence using http://www.youtube.com/watch?v=0-U4Yr9UNBo
M&M Lab for Chi Square test.
Summative and Formative Assessments (Quizzes & Tests) for each topic.
Homework and Classwork assignments based on daily lessons.
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
How can you compare the χ2 statistic to a given critical value?
TOK: Why have some research journals “banned” p -values from their articles because they deem them too misleading? In practical terms, is saying that a result is significant the same as saying it is true? How is the term “significant” used differently in different areas of knowledge?
Enrichment: When performing a χ2 test Yates continuity correction is often applied to small samples. Is it universally accepted as a valid method? In what situations would you use Yates and why? Are there other ways to deal with small sample
The χ2 goodness of fit test.
In examinations:
• the maximum number of rows or columns in a contingency table will be 4
• the degrees of freedom will always be greater than one. At SL the degrees of freedom for the goodness of fit test will always be n−1
• the χ2 critical value will be given if appropriate
• students will be expected to use technology to find a p -value and the χ2 statistic
• only questions on upper tail tests with commonly-used significance levels (1%, 5%, 10%) will be set
• students will be expected to either compare a p -value to the given significance level or compare the χ2 statistic to a given critical value
• expected frequencies will
Other contexts: Psychology: A common test is the Mann-Whitney U test. When and why is this thought to be a more reliable test in psychology?
Links to other subjects: Fieldwork (biology, psychology, environmental systems and societies, sports exercise and health science, geography).
Use of technology: Use of simulations to generate data.
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
sizes? be greater than 5.
Hand calculations of the expected values or the χ2statistic may enhance understanding.
If using χ2 tests in the Internal Assessment, students should be aware of the limitations of the test for expected frequencies of 5 or less.
The t -test.
Use of the p -value to compare the means of two populations.
Using one-tailed and two-tailed tests.
In examinations calculations will be made using technology.
Samples will be unpaired, and population variance will always be unknown.
Students will be asked to interpret the results of a test.
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NJDOE Student Learning Objective
Essential Questions Content Activities & Assessments
Resources
Students should know that the underlying distribution of the variables must be normal for the t -test to be applied.
In examinations, students should assume that variance of the two groups is equal and therefore the pooled two-sample t -test should be used.
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Unit 3 Vocabulary
PopulationSampleRandom sampleDiscrete and continuous dataFrequency distribution tablesFrequency histogramsEqual class intervals Box and whisker plotsOutliersGrouped dataMid-interval valuesInterval widthBoundariesModal classStatistical measuresMeasures of central tendency Mean (average)Median ModeBimodal QuartilePercentileDispersion (spread)Range
Interquartile range VariancePopulation variance Standard deviation (and properties of)Population standard deviation Cumulative frequency or ogive (and graphs)DataQualitative dataQuantitative dataPearson’s product moment correlation coefficientRank Spearman’s rank correlation coefficientLinearityMonotonic relationshipNull and alternative hypothesisSignificance levelp-valuet-testone-tailed and two-tailed testsexpected and observed frequencies χ2 goodness of fit and independence testscontingency tablesdegrees of freedomcritical value
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TOK Connections
Why have mathematics and statistics sometimes been treated as separate subjects? How easy is it to be misled by statistics? Is it ever justifiable to purposely use statistics to mislead others?
What is the difference between information and data? Does “data” mean the same thing in different areas of knowledge?
Could mathematics make alternative, equally true, formulae? What does this tell us about mathematical truths? Does the use of statistics lead to an over-emphasis on attributes that can be easily measured over those that cannot?
Correlation and causation–can we have knowledge of cause and effect relationships given that we can only observe correlation? What factors affect the reliability and validity of mathematical models in describing real-life phenomena?
The correlation between smoking and lung cancer was “discovered” using mathematics. Science had to justify the cause.
Does correlation imply causation? Mathematics and the world. Given that a set of data may be approximately fitted by a range of curves, where would a mathematician seek for knowledge of which equation is the “true” model?
Why have some research journals “banned” p -values from their articles because they deem them too misleading? In practical terms, is saying that a result is significant the same as saying it is true? How is the term “significant” used differently in different areas of knowledge?
Contribution to the Development of Students’ Approaches to Learning Skills
Techniques to enhance the thinking skills of the students. Ask students to formulate a reasoned argument to support their opinion or conclusion. Give students time to think through their answers before asking them for a response. Set students a task which requires higher-order thinking skills such as analysis or evaluation. Build on specific prior tasks. Help students to make their thinking more visible, for example: use a strategy such as a thinking routine.
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Problem solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics course, students will be expected to demonstrate the following:
Knowledge and understanding: Recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.
Problem solving: Recall, select and use their knowledge of mathematical skills, results and models in both abstract and real-world contexts to solve problems.
Communication and interpretation: Transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation; use appropriate notation and terminology.
Technology: Use technology accurately, appropriately and efficiently both to explore new ideas and to solve problems.
Reasoning: Construct mathematical arguments through use of precise statements, logical deduction and inference and by the manipulation of mathematical expressions.
Inquiry approaches: Investigate unfamiliar situations, both abstract and from the real world, involving organizing and analyzing information, making conjectures, drawing conclusions, and testing their validity.
Contributions to the Development of the Attribute(s) of the Learner Profile
Students exercise initiative in applying thinking skills critically and creatively to recognize and approach complex problems and make reasoned ethical decisions. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analyzing information or measurements, drawing conclusions, testing validity, and considering their scope and limitation. Reflective approaches: thoughtfully consider the world and our own ideas and experiences, extend what we learn in the classroom to life.
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The aims of all DP mathematics courses are to enable students to:1. develop a curiosity and enjoyment of mathematics, and appreciate its elegance and power2. develop an understanding of the concepts, principles and nature of mathematics3. communicate mathematics clearly, concisely and confidently in a variety of contexts4. develop logical and creative thinking, and patience and persistence in problem solving to instil confidence in using mathematics5. employ and refine their powers of abstraction and generalization6. take action to apply and transfer skills to alternative situations, to other areas of knowledge and to future developments in their local and global communities7. appreciate how developments in technology and mathematics influence each other8. appreciate the moral, social and ethical questions arising from the work of mathematicians and the applications of mathematics9. appreciate the universality of mathematics and its multicultural, international and historical perspectives10. appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course11. develop the ability to reflect critically upon their own work and the work of others12. independently and collaboratively extend their understanding of mathematics.
Contributions to the Development of International Mindedness
The physicist Frank Oppenheimer wrote: “Prediction is dependent only on the assumption that observed patterns will be repeated”. This is the danger of extrapolation. There are many examples of its failure in the past, for example share prices, the spread of disease, climate change
Discussion of the different formulae for the same statistical measure (for example, variance).
The Kinsey report–famous sampling techniques.
The benefits of sharing and analyzing data from different countries; discussion of the different formulae for variance.
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