Probability and Statistical Inference

Students use probability to make better decisions based on knowledge than on intuition alone, and use the normal distribution to understand outcomes of random processes repeated over time.

Math

Unit 8

11th Grade

Unit Summary

This unit focuses on standards in the Conditional Probability and the Rules of Probability and Making Inferences and Justifying Conclusions parts of the Statistics and Probability standards. The portion of the unit focused on probability focuses on experimental and conditional probability in an experimental context, emphasizing applications to medical testing. Probability helps us to reason about phenomena in the world and make decisions with better knowledge than relying on intuition alone. An emphasis on conditional probability helps students to reason about cause and effect and serves as an introduction to principles of experimental analysis.

The portion of the unit focused on making inferences emphasizes normal distributions and understanding the outcomes of random processes when they are repeated over time. Finally, students use distributions to make inferences about populations based on samples and apply an understanding of variability to reason about the relationship between samples and populations.

This unit is slightly abbreviated to allow our teachers time to teach the next unit, Limits and Continuity, that prepares students for calculus. Each portion of this unit addresses most but not all of the standards in their respective strands, so if teachers have extra time in the year we suggest adding lessons to extend on each topic; including but not limited to lessons that require students to model different contexts using probability, experiment using probability simulations, gather experimental data and engage with randomization, and make inferences about populations that are relevant to them.

Pacing: 15 instructional days (13 lessons, 1 flex day, 1 assessment day)

Assessment

The following assessments accompany Unit 8.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Questions of unit
- Lesson(s) that assessment points to

Internalization of Trajectory of Unit

Read and annotate “Unit Summary.”
Notice the progression of concepts through unit using the “Unit at a Glance.”
Do all target tasks. Annotate the target tasks for:
- Essential questions
- Connection to assessment questions.
Answer the essential questions.

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

Probability helps to analyze the chance of events occurring and provides a framework with which to make decisions about future events.
Statistical analysis allows us to understand populations based on information from random samples of that population.
Statistical inference is imperfect but allows us to reason using the margin of error and understand confidence in estimates to better inform decisions.

Vocabulary

Terms and notation that students learn or use in the unit

Proportion (Sample proportion/Population proportion)	Population
Tree diagram	Mutually exclusive (Disjoint)
Venn diagram	Complement
Sample space	Addition rule
Multiplication rule	Conditional probability
Relative frequency	Two-way table
Parameter	Medical testing
Characteristic	Sample
Sample survey	Experiment
Observational study	True positive/True negative
False positive/False negative

Materials

The materials, representations, and tools teachers and students will need for this unit

Spinners, dice, coins, and other tools are all useful to help students understand probability
Computer simulations (A useful and time-efficient substitute for gathering actual experimental data)
Z-score tables
(Optional) Data that is relevant to students
(Optional) Fathom (Useful for modeling using statistics)

Lesson Map

Topic A: Probability

Lesson

Determine probabilities of mutually exclusive events.

Standards

S.CP.A.1

Lesson

Determine probabilities of events that are not mutually exclusive.

Standards

S.CP.A.1S.CP.B.6S.CP.B.7

Lesson

Calculate conditional probabilities.

Standards

S.CP.A.3

Lesson

Determine when events are independent and describe independent events using everday language.

Standards

S.CP.A.2S.CP.A.3S.CP.A.5

Lesson

Calculate relative frequencies in two-way tables to analyze data and determine independence.

Standards

S.CP.A.4

Lesson

Use conditional probability to make decisions about medical testing.

Standards

S.CP.A.2S.CP.A.3

Topic B: The Normal Distribution

Lesson

Describe the center, shape, and spread of distributions by reasoning visually about the mean, standard deviation, and shape of a histogram.

Standards

S.IC.A.1

Lesson

Derive and calculate population percentages based on a normal distribution of data.

Standards

S.IC.A.2S.IC.B.4

Lesson

Use $${z-}$$scores to identify population percentiles.

Standards

S.IC.B.4

Topic C: Statistical Inferences and Conclusions

Lesson

Describe and compare statistical study methods.

Standards

S.IC.B.3S.IC.B.6

Lesson

Use multiple random samples to estimate a population mean or proportion and verify the validity of the sampling method by analyzing the means and standard errors of samples.

Standards

S.IC.B.4

Lesson

Calculate and describe the margin of error in context and use larger sample sizes to minimize the margin of error.

Standards

S.IC.B.4

Lesson

Compare two treatments in experimental data and determine if the difference between the two treatments is significant.

Standards

S.IC.B.5

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

High School — Statistics and Probability

High School — Statistics and Probability

S.CP.A.1 — Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
High School — Statistics and Probability

S.CP.A.2 — Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
High School — Statistics and Probability

S.CP.A.3 — Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
High School — Statistics and Probability

S.CP.A.4 — Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
High School — Statistics and Probability

S.CP.A.5 — Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
High School — Statistics and Probability

S.CP.B.6 — Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
High School — Statistics and Probability

S.CP.B.7 — Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
High School — Statistics and Probability

S.IC.A.1 — Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
High School — Statistics and Probability

S.IC.A.2 — Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
High School — Statistics and Probability

S.IC.B.3 — Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
High School — Statistics and Probability

S.IC.B.4 — Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
High School — Statistics and Probability

S.IC.B.5 — Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
High School — Statistics and Probability

S.IC.B.6 — Evaluate reports based on data.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

High School — Statistics and Probability

High School — Statistics and Probability

S.ID.A.4 — Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
High School — Statistics and Probability

S.ID.B.5 — Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Statistics and Probability

Statistics and Probability

7.SP.A.1 — Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Statistics and Probability

7.SP.B.3 — Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
Statistics and Probability

7.SP.B.4 — Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Statistics and Probability

7.SP.C.5 — Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Statistics and Probability

7.SP.C.6 — Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Statistics and Probability

7.SP.C.7 — Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Statistics and Probability

7.SP.C.8 — Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Future Standards

Standards in future grades or units that connect to the content in this unit

High School — Statistics and Probability

High School — Statistics and Probability

S.CP.B.8 — Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
High School — Statistics and Probability

S.CP.B.9 — Use permutations and combinations to compute probabilities of compound events and solve problems.
High School — Statistics and Probability

S.MD.A.1 — Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
High School — Statistics and Probability

S.MD.A.2 — Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
High School — Statistics and Probability

S.MD.A.3 — Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
High School — Statistics and Probability

S.MD.A.4 — Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

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