Probability and Statistics Curriculum - Overview Probability and Statistics is a full year study designed primarily as a preparation course for college, technical school or junior college. The key components in probability are probability terms, the concept of the probability of an event, predicting and determining probabilities, expected value, the relationship between theoretical and experimental probabilities, and compound events. In statistics, the key components are data collection, organization, representation, sampling, central tendency, variance and correlation, and analysis and inference. Probability and Statistics are the mathematics used to understand chance and to collect, organize, describe, and analyze numerical data. From weather reports to sophisticated studies of genetics, from election results to product preference survey, probability and statistical language and concepts are increasingly present in the media and in everyday conversations. Students need this mathematics to help them judge the correctness of an argument supported by seemingly persuasive data. Course topics will include the study of introduction to statistics, summarizing and graphing data, statistics for describing, exploring, and comparing data, probability, discrete probability distributions, normal probability distributions, estimates and sample sizes, hypothesis testing, inferences from two samples, and correlation and regression. Graphing calculators, Excel, GeoGebra and real life applications are used throughout the course to develop conceptual understanding and analysis of data. By the end of the course students will be sensible, critical users of probability and statistics, able to apply the processes and principles developed in this course to real-world problems. Students should not think that those people who did not win the lottery yesterday have a greater chance of winning today! They should not believe an argument merely because various statistics are offered. Rather, they should be able to judge whether the statistics are meaningful and are being used appropriately.

Curriculum – Standards Probability & Statistics

Unit 1 Unit 2 Unit 3 Unit 4

S-IC3 S-ID1 S-ID2 S-ID3 S-ID5

S-CP1 S-CP2 S-CP3 S-CP4 S-CP5 S-CP6 S-CP7 S-CP8 S-CP9 S-MD1 S-MD2 S-MD3 S-MD4 S-MD5

S-ID4 S-MD1 S-ID6 S-IC1 S-IC3 S-IC4

S-MD7 S-IC4 S-IC5 S-IC6 S-ID6 S-ID7 S-ID8 S-ID9

PCTI MATHEMATICS DEPARTMENT

Probability and Statistics UNIT 1 • Introduction to Statistics • Summarizing and Graphing Data • Statistics for Describing, Exploring, and Comparing Data

TECHNOLOGY STANDARDS KEY VOCABULARY

Use a graphing calculator/geogebra/ videos to: • Graphs

https://www.youtube.com/results?search_query=pareto+chart+statistics https://www.youtube.com/watch?v=qHW3bo5SH0g https://www.youtube.com/watch?v=qHW3bo5SH0g https://www.khanacademy.org/math/pre-algebra/applying-math-reasoning-topic/reading_data/v/u08-l1-t2-we3-stem-and-leaf-plots https://www.youtube.com/results?search_query=pie+chart https://www.youtube.com/results?search_query=pareto+chart+statistics http://tube.geogebra.org/search/results/uid/3AFIIbWBcm https://www.youtube.com/results?search_query=graph+historam+with+geogebra

• Descriptive Statistics https://www.khanacademy.org/math/probability/descriptive-statistics https://www.youtube.com/results?search_query=find+mean+median+mode+range+with+ti+84 https://www.youtube.com/results?search_query=find+mean+median+mode+range+with+ti+nspire https://www.youtube.com/results?search_query=find+boxplot+with+ti+nspire

Data Statistics Population Census Sample Parameter Quantitative Categorical Discrete data Continuous (numerical) data Nominal level of measurement Ordinal level of measurement Interval level of measurement Voluntary response sample

Frequency distribution Lower class limits Upper class limits Class boundaries Class midpoints Class width Histogram Relative frequency histogram Frequency polygon Ogive Dotplots Stemplots Bar graphs

Observational study Experiment Simple random sample Random sample Probability sample Systematic sampling Convenience sampling Stratified sampling Clustering sampling Cross-sectional study Retrospective study Confounding Sampling error Outliers Percentiles

Pareto charts Pies charts Scatterplots Time-Series Graph Measure of Center Arithmetic mean Median, Mode, Midrange Weighted mean Skewed Symmetric Range Standard Deviation Variance Coefficient of variation Z score Quartiles Boxplot

# TOPICS (textbook reference; # of days for instruction)

# STUDENT LEARNING OBJECTIVES CCSS code

I INTRODUCTION TO STATISTICS (1.1-1.5 ; 10 days)

1 day 1 Review and Preview 2 days 2 Statistical Thinking S-IC3 2 days 3 Types of Data S-IC3 2 days 4 Critical Thinking S-IC3 3 days 5 Collecting Sample Data S-IC3 II SUMMARIZING AND GRAPHIND DATA

(2.1-2.5 ; 15 days)

1 day 1 Review and Preview 4 days 2 Frequency Distributions S-ID5 4 days 3 Histograms S-ID1, S-ID5 5 days 4 Statistical Graphics S-ID1, S-ID2 2 days 5 Critical Thinking: Bad Graphs S-ID1, S-ID2 lll STATISTICS FOR DESCRIBING, EXPLORING, AND

COMPARING DATA (3.1-3.4 ; 15 days)

1 day 1 Review and Preview 6 days 2 Measures of Center S-ID2, S-ID3 5 days 3 Measures of Variation S-ID2, S-ID3 3 days 4 Measures of Relative Standing and Boxplots S-ID1

Selected Opportunities for Connections to Mathematical Practices

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of

others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Code # Common Core State Standards

S-IC3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

S-ID1 Represent data with plots on the real number line (dot plots, histograms, and box plots). S-ID2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile

range, standard deviation) of two or more different data sets. S-ID3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data

points (outliers). S-ID5 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.

PCTI MATHEMATICS DEPARTMENT

Probability and Statistics UNIT 2 • Probability • Discrete Probability Distributions

TECHNOLOGY STANDARDS KEY VOCABULARY

Use a graphing calculator/geogebra/ videos to: • Find probability.

https://www.youtube.com/results?search_query=find+probability+addition+rule https://www.youtube.com/results?search_query=find+probability+multiplication+rule https://www.youtube.com/results?search_query=find+complements+and+conditional+probability https://www.youtube.com/results?search_query=probability+through+simulation https://www.youtube.com/results?search_query=counting+principle https://www.youtube.com/results?search_query=permutation+and+combination https://www.youtube.com/results?search_query=permutation+and+combination+using+calculator https://www.youtube.com/results?search_query=permutation+and+combination+using+ti+nspire

• Find Probability distributions. https://www.youtube.com/results?search_query=random+variables https://www.youtube.com/results?search_query=binomial+probability+distribution https://www.youtube.com/results?search_query=mean+variance+standard+deviation+binomial+probability+distribution https://www.youtube.com/results?search_query=poisson+distribution http://www.geogebratube.org/search/results/uid/4CEjGqE0H7

Event Simple Event Sample Space Complement Actual Odds Against Actuals Odds in Favor Payoff Odds Compound Event Disjoint Independent Conditional Probability Simulation

Random Variable Probability Distribution Discrete Random Variable Continuous Random Variable Expected Value Binomial Probability Distribution Poisson Distribution

http://www.geogebratube.org/search/results/uid/EPS7dvERw2

# TOPICS (textbook reference; # of days for instruction)

# STUDENT LEARNING OBJECTIVES CCSS code

IV PROBABILITY (4.1-4.8 ; 20 days)

1 day 1 Review and Preview 4 days 2 Basic Concepts of Probability S-CP1-2, S-MD6, 3 days 3 Addition Rule S-CP1, S-CP7, S-MD4 3 days 4 Multiplication Rule: Basics S-CP2, S-CP6 3 days 5 Multiplication Rule: Complements and Conditional

Probability S-CP3, S-CP6, S-CP8

3 days 6 Probabilities through Simulations S-CP4, S-CP5 3 days 7 Counting S-CP9

V DISCRETE PROBABILITY DISTRIBUTIONS (5.1-5.5 ; 20 days)

1 day 1 Review and Preview 6 days 2 Random Variables S-MD1-5 5 days 3 Binomial Probability Distributions S-MD1 5 days 4 Mean, Variance, and Standard Deviation for the Binomial

Distribution S-MD2, S-ID2-3

Selected Opportunities for Connections to Mathematical Practices

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of

others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Code # Common Core State Standards

S-CP1 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-CP2 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-CP3 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-CP4 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-CP5 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. Use the rules of probability to compute probabilities of compound

S-CP6 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-CP7 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-CP8 (+) 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. S-CP9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. S-MD1 (+) 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. S-MD2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. S-MD3 (+) 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.

S-MD4 (+) 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?

S-MD5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fastfood restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

PCTI MATHEMATICS DEPARTMENT

Probability and Statistics UNIT 3 • Normal Probability Distributions • Estimates and Sample Sizes

TECHNOLOGY STANDARDS KEY VOCABULARY

Use a graphing calculator/geogebra/ videos to:

• Find normal probability Distributions. https://www.youtube.com/results?search_query=stanard+normal+distribution https://www.youtube.com/results?search_query=appliction+of+normal+distribution https://www.youtube.com/results?search_query=central+limit+theorem https://www.youtube.com/results?search_query=normal+as+an+approximation+to+binomial https://www.youtube.com/results?search_query=assessing+normaility+in+statistics http://www.geogebratube.org/search/results/uid/ZUIZkefdAN

• Estimate and find sample size for population parameters https://www.youtube.com/results?search_query=estimate+population+proportion https://www.youtube.com/results?search_query=estimate+population+mean https://www.youtube.com/results?search_query=estimate+population+variance+statistics

Normal Distribution Uniform Distribution Standard Normal Distribution Sampling Distribution of a Statistics Sampling Distribution of the Mean Sampling Distribution of the Variance Sampling Distribution of the Proportion Continuity Correction Normal Quantile Plot

Point Estimate Confidence Interval Confidence Level Critical Value Margin of Error Degrees of Freedom

# TOPICS (textbook reference; # of days for instruction)

# STUDENT LEARNING OBJECTIVES CCSS code

VI NORMAL PROBABILITY DISTRIBUTIONS (6.1-6.7 ; 25 days)

1 day 1 Review and Preview 4 days 2 The Standard Normal Distribution S-ID4 4 days 3 Applications of Normal Distribution S-ID4 4 days 4 Sampling Distributions and Estimators S-ID4 4 days 5 The Central Limit Theorem S-ID4 4 days 6 Normal as Approximation to Binomial S-ID4, S-MD1 4 days 7 Determining Normality S-ID6

VII ESTIMATES AND SAMPLE SIZES (7.1-7.5 ; 15 days)

1 day 1 Review and Preview 5 days 2 Estimating a Population Proportion. S-IC1, S-IC3, S-IC4, S-

ID4 3 days 3 Estimating a Population Mean: δ Known. S-IC1, S-IC3, S-IC4 3 days 4 Estimating a Population Mean: δ Not Known S-IC1, S-IC3, S-IC4 3 days 5 Estimating a Population Variance S-IC1, S-IC3, S-IC4

Selected Opportunities for Connections to Mathematical Practices

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of

others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Code # Common Core State Standards

S-ID4 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.

S-MD1 (+) 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.

S-ID6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals.

c. Fit a linear function for a scatter plot that suggests a linear association. S-IC1 Understand statistics as a process for making inferences about population parameters based on a random sample from that

population. S-IC3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how

randomization relates to each. S-IC4 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.

PCTI MATHEMATICS DEPARTMENT

Probability and Statistics UNIT 4 • Hypothesis Testing • Inferences from two samples • Correlation and Regression

•

TECHNOLOGY STANDARDS KEY VOCABULARY

Use a graphing calculator/geogebra/ videos to:

• Test a claim https://www.youtube.com/results?search_query=hypothesis+testing+statistics https://www.youtube.com/results?search_query=test+claim+about+proportion https://www.youtube.com/results?search_query=test+claim+about+mean https://www.youtube.com/results?search_query=test+claim+about+variation http://www.geogebratube.org/search/results/uid/6Rm5JpOUmP

• Test a claim about two samples

https://www.youtube.com/results?search_query=inferences+about+two+proportions https://www.youtube.com/results?search_query=inferences+about+two+means https://www.youtube.com/results?search_query=comparing+variation+in+two+samples

• Find correlation and regression

Hypothesis Hypothesis Test Independent Dependent

Correlation Linear Correlation Coefficient Regression Equation Regression Line Marginal Change Outlier Influential Points Residual Least-squares property Residual Plot Total deviation Explained Deviation Unexplained Deviation Coefficient of Determination Prediction Interval Multiple Regression Equation

https://www.youtube.com/results?search_query=correlation+and+regression+in+statistics https://www.youtube.com/results?search_query=correlation+and+regression+ti+nspire http://www.geogebratube.org/search/results/uid/8oQC2sCx6D

Adjusted Coefficient of Determination

# TOPICS (textbook reference; # of days for instruction)

# STUDENT LEARNING OBJECTIVES CCSS code

VIII HYPOTHESIS TESTING (8.1-8.6 ; 15 days)

1 day 1 Review and Preview 4 days 2 Basics of Hypothesis Testing S-MD7 4 days 3 Test a Claim about a Proportion S-IC4 3 days 4 Test a Claim about a Mean: δ Known S-IC4 2 days 5 Test a Claim about a Mean: δ Not Known S-IC4 2 days 6 Test a Claim about Variation S-IC4 IX INFERENCES FROM TWO SAMPLES

(9.1-9.5 ; 15 days)

1 day 1 Review and Preview 5 days 2 Inferences about Two Proportions S-IC4, S-IC5, S-IC6 3 days 3 Inferences about Two Means; Independent Samples S-IC4, S-IC5, S-IC6 3 days 4 Inferences from Matched Pairs S-IC4, S-IC5, S-IC6 3 days 5 Comparing Variation in Two Samples S-IC4, S-IC5, S-IC6

X CORRELATION AND REGRESSION

(10.1-10.6 ; 10 days)

1 day 1 Review and Preview 1 day 2 Correlation S-ID6 2 days 3 Regression S-ID6, S-ID8 2 days 4 Variation and Prediction Intervals S-ID9 2 days 5 Multiple Regressions S-ID8, S-ID9 2 days 6 Modeling S-ID7

Selected Opportunities for Connections to Mathematical Practices

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of

others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Code # Common Core State Standards

S-MD7 Analyze decisions and strategies using probability concepts ( eg. Product testing, medical testing, pulling a hockey goalies at the end of a game ).

S-IC4 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-IC5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

S-IC6 Evaluate reports based on data. S-ID6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association.

S-ID7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S-ID8 Compute (using technology) and interpret the correlation coefficient of a linear fit. S-ID9 Distinguish between correlation and causation.