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.
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
This assessment accompanies Unit 8 and should be given on the suggested assessment day or after completing the unit.
Unit Prep
Intellectual Prep
?
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
?
- 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.
Materials
?
- 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)
Vocabulary
?
| 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 |
Lesson Map
Topic A: Probability
1
S.CP.A.1
Determine probabilities of mutually exclusive events.
2
S.CP.A.1
S.CP.B.6
S.CP.B.7
Determine probabilities of events that are not mutually exclusive.
3
S.CP.A.3
Calculate conditional probabilities.
4
S.CP.A.2
S.CP.A.3
S.CP.A.5
Determine when events are independent and describe independent events using everday language.
5
S.CP.A.4
Calculate relative frequencies in two-way tables to analyze data and determine independence.
6
S.CP.A.2
S.CP.A.3
Use conditional probability to make decisions about medical testing.
Topic B: The Normal Distribution
7
S.IC.A.1
Describe the center, shape, and spread of distributions by reasoning visually about the mean, standard deviation, and shape of a histogram.
8
S.IC.A.2
S.IC.B.4
Derive and calculate population percentages based on a normal distribution of data.
9
S.IC.B.4
Use $${z-}$$scores to identify population percentiles.
Topic C: Statistical Inferences and Conclusions
10
S.IC.B.3
S.IC.B.6
Describe and compare statistical study methods.
11
S.IC.B.4
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.
12
S.IC.B.4
Calculate and describe the margin of error in context and use larger sample sizes to minimize the margin of error.
13
S.IC.B.5
Compare two treatments in experimental data and determine if the difference between the two treatments is significant.
Common Core Standards
Key: Major Cluster Supporting Cluster Additional Cluster
Core Standards
?
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").
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
S.IC.A.1 — Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
-
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?
-
S.IC.B.3 — Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
-
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.
-
S.IC.B.5 — Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
-
S.IC.B.6 — Evaluate reports based on data.
Foundational Standards
?
High School — Statistics and Probability
-
S.ID.A.4
-
S.ID.B.5
Statistics and Probability
-
7.SP.A.1
-
7.SP.B.3
-
7.SP.B.4
-
7.SP.C.5
-
7.SP.C.6
-
7.SP.C.7
-
7.SP.C.8
Future Standards
?
High School — Statistics and Probability
-
S.CP.B.8
-
S.CP.B.9
-
S.MD.A.1
-
S.MD.A.2
-
S.MD.A.3
-
S.MD.A.4
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.