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Part 1: Introduction to Statistics

Pacing: 1 semester

Mathematical Practices

Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning. Practices in bold are to be emphasized in the unit. 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.

Standards Overview

Summarize, represent, and interpret data on a single count or measurement variable Summarize, represent, and interpret data on quantitative variables. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points.

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Part 1: Introduction to Statistics

Priority and Supporting CCSS Explanations and Examples*

CC.9-12.S.ID.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.

Students may use spreadsheets, graphing calculators, statistical software and tables to analyze the fit between a data set and normal distributions and estimate areas under the curve. Examples:

● The bar graph below gives the birth weight of a population of 100

chimpanzees. The line shows how the weights are normally

distributed about the mean, 3250 grams. Estimate the percent of

baby chimps weighing 3000-3999 grams.

● Determine which situation(s) is best modeled by a normal distribution. Explain your reasoning. o Annual income of a household in the U.S. o Weight of babies born in one year in the U.S.

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Part 1: Introduction to Statistics

Priority and Supporting CCSS Explanations and Examples*

CC.9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

CC.9-12.S.IC.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?

Possible data-generating processes include (but are not limited to): flipping coins, spinning spinners, rolling a number cube, and simulations using the random number generators. Students may use graphing calculators, spreadsheet programs, or applets to conduct simulations and quickly perform large numbers of trials. The law of large numbers states that as the sample size increases, the experimental probability will approach the theoretical probability. Comparison of data from repetitions of the same experiment is part of the model building verification process. Example:

● Have multiple groups flip coins. One group flips a coin 5 times, one

group flips a coin 20 times, and one group flips a coin 100 times.

Which group’s results will most likely approach the theoretical

probability?

CC.9-12.S.IC.6 Evaluate reports based on data. Explanations can include but are not limited to sample size, biased survey sample, interval scale, unlabeled scale, uneven scale, and outliers that distort the line-of-best-fit. In a pictogram the symbol scale used can also be a source of distortion. As a strategy, collect reports published in the media and ask students to consider the source of the data, the design of the study, and the way the data are analyzed and displayed.

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Part 1: Introduction to Statistics

Priority and Supporting CCSS Explanations and Examples*

Example: ● A reporter used the two data sets below to calculate the mean

housing price in Arizona as $629,000. Why is this calculation not

representative of the typical housing price in Arizona?

o King River area {1.2 million, 242000, 265500, 140000, 281000,

265000, 211000}

o Toby Ranch homes {5million, 154000, 250000, 250000,

200000, 160000, 190000}

S-ID 1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

Students may compare and contrast the advantage of each of these representations.

S-ID 2. 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.

Students may use spreadsheets, graphing calculators and statistical software for calculations, summaries, and comparisons of data sets. Example: Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, median and standard deviation. Explain how the values vary about the mean and median. What information does this give the teacher?

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Part 1: Introduction to Statistics

Priority and Supporting CCSS Explanations and Examples*

S-ID 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Example: After the 2009-2010 NBA season LeBron James switched teams from the Cleveland Cavaliers to the Miami Heat, and he remained the top scorer (in points per game) in his first year in Miami. Compare team statistics for Cleveland (2009-2010) and Miami (2010-2011) for all players who averaged at least 10 minutes per game. Using the 1.5 X IQR rule, determine for which team and year James’s performance may be considered an outlier.

Concepts What Students Need to Know

Skills What Students Need To Be Able To Do

Bloom’s Taxonomy Levels

Data (Who, What, When, Where, Why)

Measures of central tendency (mean, median, mode)

Measures of spread (range, interquartile range, standard deviation)

Outlier

Histogram

Box plot

Normal Distribution

Identify

Interpret

Classify (Categorical, Quantitative)

Calculate (mean, median, mode, interquartile range)

Use (statistics to infer)

Use (technology to find standard deviation)

Interpret (correlation)

Identify (outliers)

Fit (normal distribution)

1 2 1

1

3 3

2 4

3

3

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Part 1: Introduction to Statistics

Concepts What Students Need to Know

Skills What Students Need To Be Able To Do

Bloom’s Taxonomy Levels

Population percentages

Appropriate use of normal approx

Calculators, spreadsheets, tables

Estimate (population percentages)

Recognize (inappropriate usage)

Use (technology)

Estimate (area under normal)

4

3

3

4

Essential Questions

How can the properties of data be communicated to illuminate its important features?

Corresponding Big Ideas

Statisticians summarize, represent , and interpret categorical and quantitative data in multiple ways since one method can reveal or create a different impression than another.

Standardized Assessment Correlations (State, College and Career)

Expectations for Learning (in development)

CollegeBoard PSAT and SAT

Vocabulary

Population, sample, variable, quantitative variable, categorical variable, relative frequency, contingency table, marginal distribution, conditional distribution, histogram, symmetrical data, skewed data, outlier value, mean, median, interquartile range, standard deviation, Normal model, z-score

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Part 1: Introduction to Statistics

Learning Activities

Topic Source(s) CCSS

Define Data – 5 W’s Categorical Data Create a relative frequency table Analyze data displays Calculate marginal distributions from a contingency table Calculate conditional distribution from a contingency table Determine whether variables are dependent/independent Quantitative Data Create histogram by hand and with calculator Discuss shape, center, spread Calculate range, median, quartiles and interquartile range Calculate population percentages based on quartile info Mean, Median, Standard Deviation, IQR (Interquartile range) Calculate Determine most appropriate measure based on data info Normal Distribution Determine whether a data set is normal Z-Scores Calculate Use scores to compare data

Stats in Your World text CC.9-12.S.IC.1 CC.9-12.S.IC.2

S-ID5

CC.9-12.S.IC.6 S-ID 3

S-ID 1 S-ID 2 S-ID 3

CC.9-12.S.ID.4 S-ID 2 S-ID 3

CC.9-12.S.ID.4

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Part 1: Introduction to Statistics

Unit Assessments The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.

Section quizzes, SAT released items, end of unit assessment

Application

The Pew Research Center for the People and the Press (http://people-press.org) has asked a representative sample of U.S. adults about global

warming, repeating the question over time. In January, 2007 the responses reflected an increased belief that global warming is real and due to

human activity. Below is a display of the percentages of respondents choosing each of the major

alternatives offered, list two errors in the display.

Normal Model 68-95-99.7 model Sketch with deviations identified Calculate percentages of a population Calculate z-scores for a given population percentage

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Part 1: Introduction to Statistics

A town’s January high temperatures average 36oF with a standard deviation of 8

o, while in July the mean high temperature is 75

oF and the

standard deviation is 10o. In which month is it more unusual to have a day with a high temperature of 55

o? Show work and explain your answer.

Companies who design furniture for elementary school classrooms produce a variety of sizes for kids of different ages. Suppose the heights of

kindergarten children can be described by a Normal model with a mean of 39.2 inches and a standard deviation of 1.9 inches.

a. Calculate the z-score for a child’s height of 42 inches.

b. Calculate the z-score for a child’s height of 35 inches.

c. Use your calculator to determine the percent of children whose heights fall 35 and 42 inches. Write down the calculator “operation” as

your work.

d. What percent of kindergarten children should the company expect to be less than 35 inches tall? Again use your calculator and write

down the “operation” as your work.

e. Use your calculator and find the z-score for the lowest 20% of the children’s heights. Find the height that corresponds to this z-score,

then sketch and shade in the normal curve showing this 20%. Clearly label your graph.

f. Use your calculator and find the z-score for the highest 4% of the children’s heights. Find the height that corresponds to this z-score,

then sketch and shade in the normal curve showing this 4%. Clearly label your graph.

Draft 1

Part 2: Inferential Statistics

Pacing: 1 semester

Mathematical Practices

Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning. Practices in bold are to be emphasized in the unit. 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.

Standards Overview

Use the mean and standard deviation of a data set to fit it to a normal distrubtion and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. 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. Use data from a randomized experiment to compare two treatments; use simulations to deide if differences between parameters are significant.

Priority and Supporting CCSS Explanations and Examples*

CC.9-12.S.ID.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.

Students may use spreadsheets, graphing calculators, statistical software and tables to analyze the fit between a data set and normal distributions and estimate areas under the curve. Examples:

● The bar graph below gives the birth weight of a population of 100

Draft 1

Part 2: Inferential Statistics

Priority and Supporting CCSS Explanations and Examples*

chimpanzees. The line shows how the weights are normally

distributed about the mean, 3250 grams. Estimate the percent of

baby chimps weighing 3000-3999 grams.

● Determine which situation(s) is best modeled by a normal

distribution. Explain your reasoning.

o Annual income of a household in the U.S.

o Weight of babies born in one year in the U.S.

Draft 1

Part 2: Inferential Statistics

Priority and Supporting CCSS Explanations and Examples*

CC.9-12.S.IC.1 Understand statistics as a process for

making inferences about population parameters based

on a random sample from that population.

CC.9-12.S.IC.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?

Possible data-generating processes include (but are not limited to): flipping coins, spinning spinners, rolling a number cube, and simulations using the random number generators. Students may use graphing calculators, spreadsheet programs, or applets to conduct simulations and quickly perform large numbers of trials. The law of large numbers states that as the sample size increases, the experimental probability will approach the theoretical probability. Comparison of data from repetitions of the same experiment is part of the model building verification process. Example:

● Have multiple groups flip coins. One group flips a coin 5 times, one

group flips a coin 20 times, and one group flips a coin 100 times.

Which group’s results will most likely approach the theoretical

probability?

CC.9-12.S.IC.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.

Students may use computer generated simulation models based upon sample surveys results to estimate population statistics and margins of error.

Draft 1

Part 2: Inferential Statistics

Priority and Supporting CCSS Explanations and Examples*

CC.9-12.S.IC.3 Recognize the purposes of and differences

among sample surveys, experiments, and observational

studies; explain how randomization relates to each.

Students should be able to explain techniques/applications for randomly selecting study subjects from a population and how those techniques/applications differ from those used to randomly assign existing subjects to control groups or experimental groups in a statistical experiment. In statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator (for example, observing data on academic achievement and socio-economic status to see if there is a relationship between them). This is in contrast to controlled experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group before the start of the treatment.

CC.9-12.S.IC.5 Use data from a randomized experiment

to compare two treatments; use simulations to decide if

differences between parameters are significant.

Students may use computer generated simulation models to decide how likely it is that observed differences in a randomized experiment are due to chance. Treatment is a term used in the context of an experimental design to refer to any prescribed combination of values of explanatory variables. For example, one wants to determine the effectiveness of weed killer. Two equal parcels of land in a neighborhood are treated; one with a placebo and one with weed killer to determine whether there is a significant difference in effectiveness in eliminating weeds.

Draft 1

Part 2: Inferential Statistics

Priority and Supporting CCSS Explanations and Examples*

CC.9-12.S.IC.6 Evaluate reports based on data. Explanations can include but are not limited to sample size, biased survey sample, interval scale, unlabeled scale, uneven scale, and outliers that distort the line-of-best-fit. In a pictogram the symbol scale used can also be a source of distortion. As a strategy, collect reports published in the media and ask students to consider the source of the data, the design of the study, and the way the data are analyzed and displayed. Example:

● A reporter used the two data sets below to calculate the mean

housing price in Arizona as $629,000. Why is this calculation not

representative of the typical housing price in Arizona?

o King River area {1.2 million, 242000, 265500, 140000, 281000,

265000, 211000}

o Toby Ranch homes {5million, 154000, 250000, 250000,

200000, 160000, 190000}

Draft 1

Part 2: Inferential Statistics

Concepts What Students Need to Know

Skills What Students Need To Be Able To Do

Bloom’s Taxonomy Levels

● Mean and standard deviation

● Normal distribution

● Population percentages

● Appropriate use of normal approx

● Calculators, spreadsheets, tables

● Area under normal curve

● Use (statistics to infer)

● Fit (normal distribution)

● Estimate (population percentages)

● Recognize (inappropriate usage)

● Use (technology)

● Estimate (area under normal)

3

3

3

4

3

3

● Inference

● Population parameters

● Random samples

● Make (inferences based on data) 4

● Sample survey data

● Population mean / proportion

● Margin of error

● Simulation models

● Random sampling

● Use (data to make inference)

● Estimate (population parameters)

● Develop (margin of error)

4

3

3

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Part 2: Inferential Statistics

Concepts What Students Need to Know

Skills What Students Need To Be Able To Do

Bloom’s Taxonomy Levels

● Randomized experiment data

● Two treatments

● Simulations

● Significant differences in parameters

● Use (randomization process)

● Compare (treatment groups)

● Use (simulations to make decisions)

4

2

5

Essential Questions

How can a population be described when it is so large, it would be nearly impossible to collect all of the data?

Corresponding Big Ideas

Statisticians design experiments based on random samples and analyze the data to estimate the important properties of a population and make informed judgments.

Standardized Assessment Correlations (State, College and Career)

Expectations for Learning (in development)

CollegeBoard PSAT and SAT

Draft 1

Part 2: Inferential Statistics

Unit Vocabulary

Inference, population parameter, random sample, population, statistics, population mean, sample mean, population proportion, sample proportion, sample survey, margin of error, simulation model, random sampling, confidence interval,

Unit Assessments The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.

Section quizzes End-of-unit test

Learning Activities

Topic Section in Text CCSS