Classroom Resources (11) |

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 41 :

[MA2015] AL1 (9-12) 42 :

[MA2015] AL1 (9-12) 43 :

[MA2015] PRE (9-12) 45 :

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 49 :

[MA2015] PRE (9-12) 40 :

[MA2015] PRE (9-12) 39 :

[MA2015] PRE (9-12) 41 :

41 ) Represent data with plots on the real number line (dot plots, histograms, and box plots). [S-ID1]

[MA2015] AL1 (9-12) 42 :

42 ) 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-ID2]

[MA2015] AL1 (9-12) 43 :

43 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [S-ID3]

[MA2015] PRE (9-12) 45 :

45 ) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2]

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'

[MA2015] PRE (9-12) 46 :

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

[MA2015] PRE (9-12) 49 :

49 ) Evaluate reports based on data. [S-IC6]

[MA2015] PRE (9-12) 40 :

40 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (*Identify unifrom, skewed, and normal distridutions in a set of data. Determine the quartiles and interquartile range for a set of data.*) [S-ID3] (Alabama)

[MA2015] PRE (9-12) 39 :

39 ) 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. (*Focus on increasing rigor using standard deviation*). [S-ID2] (Alabama)

[MA2015] PRE (9-12) 41 :

41 ) 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-ID4]

One way we analyze data is to look at measures of central tendency—mean, median, and mode. They are the tools to look at the information for the purpose of answering the question, “What is normal?” Understanding the measures of central tendency can help us make important life decisions. For example, averages can help us set goals or plan budgets. At the end of this lesson about central tendency, students will be able to recognize and apply the concepts of mean, median, and mode in real-life problems.

View Standards
**Standard(s): **
[MA2015] PRE (9-12) 45 :

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 49 :

45 ) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2]

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'

[MA2015] PRE (9-12) 46 :

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

[MA2015] PRE (9-12) 49 :

49 ) Evaluate reports based on data. [S-IC6]

Students analyze economic data to better understand America’s middle class, its role in the economy, and its impact on economic growth. In this interactive lesson, students use media produced for How the Deck Is Stacked, and tables and graphs created from Pew Research Center and government data to examine household income and spending trends and the widening wealth gap between the upper and lower-income tiers. Interim assessments evaluate students’ ability to interpret data, make inferences, and justify conclusions. At the end of the lesson, students write an evidence-based essay on why a shrinking middle-class matters to the U.S. economy.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 41 :

[MA2015] AL1 (9-12) 42 :

[MA2015] AL1 (9-12) 43 :

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 40 :

[MA2015] PRE (9-12) 39 :

[MA2015] PRE (9-12) 41 :

41 ) Represent data with plots on the real number line (dot plots, histograms, and box plots). [S-ID1]

[MA2015] AL1 (9-12) 42 :

42 ) 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-ID2]

[MA2015] AL1 (9-12) 43 :

43 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [S-ID3]

[MA2015] PRE (9-12) 46 :

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

[MA2015] PRE (9-12) 40 :

40 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (*Identify unifrom, skewed, and normal distridutions in a set of data. Determine the quartiles and interquartile range for a set of data.*) [S-ID3] (Alabama)

[MA2015] PRE (9-12) 39 :

39 ) 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. (*Focus on increasing rigor using standard deviation*). [S-ID2] (Alabama)

[MA2015] PRE (9-12) 41 :

41 ) 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-ID4]

Introduce high school students to the art and science of statistics in the 6-minute video, "What is Statistics?" from the Against All Odds series. This video resource will demonstrate how gathering, organizing, drawing, and analyzing data is applicable in everyday life and a variety of careers.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 42 :

[MA2015] AL1 (9-12) 43 :

[MA2015] PRE (9-12) 45 :

[MA2015] PRE (9-12) 46 :46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2015] PRE (9-12) 47 :

[MA2015] PRE (9-12) 48 :

[MA2015] PRE (9-12) 40 :

[MA2015] PRE (9-12) 39 :

[MA2015] PRE (9-12) 41 :

42 ) 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-ID2]

[MA2015] AL1 (9-12) 43 :

43 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [S-ID3]

[MA2015] PRE (9-12) 45 :

45 ) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2]

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'

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 47 :

47 ) 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-IC4]

[MA2015] PRE (9-12) 48 :

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

[MA2015] PRE (9-12) 40 :

40 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (*Identify unifrom, skewed, and normal distridutions in a set of data. Determine the quartiles and interquartile range for a set of data.*) [S-ID3] (Alabama)

[MA2015] PRE (9-12) 39 :

39 ) 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. (*Focus on increasing rigor using standard deviation*). [S-ID2] (Alabama)

[MA2015] PRE (9-12) 41 :

41 ) 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-ID4]

This exercise was developed to complement the film *The National Parks of Texas* by Texas PBS & Villita Media. In this activity, students will learn about estimating the number of trees in a large area based on a smaller area.

This is one way statisticians measure forests and other wide expanses of land. It's also a great way to illustrate how polling works. Scientists will interview a smaller sample size of Americans, rather than every single American, and then make estimations based on their results. In the same way, we counted smaller samples of trees, rather than all of the trees individually to get an estimate of how many trees are in the park total.

**Note: The corresponding lesson plan can be found under the "Support Materials for Teachers" link on the right side of the page. **

View Standards
**Standard(s): **
[MA2015] PRE (9-12) 46 : 46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

Explore real-life applications for statistical data sets and creating stemplots in the 12-minute program "Stemplots" from the Against All Odds series. Students will learn how designers and engineers calculate uniform sizing for the military by creating a stemplot to visualize the data. A stemplot demonstrates how a data collection of measurements helps the military design gear that fits modern troops.

View Standards
**Standard(s): **
[MA2015] (7) 18 :

[MA2015] PRE (9-12) 45 :45 ) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2]

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'

[MA2015] PRE (9-12) 46 :46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2015] PRE (9-12) 48 :

[MA2015] PRE (9-12) 49 :

[MA2019] REG-7 (7) 10 :

18 ) Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7-SP2]

Example: Estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

[MA2015] PRE (9-12) 45 :

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 48 :

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

[MA2015] PRE (9-12) 49 :

49 ) Evaluate reports based on data. [S-IC6]

[MA2019] REG-7 (7) 10 :

10. Examine a sample of a population to generalize information about the population.

a. Differentiate between a sample and a population.

b. Compare sampling techniques to determine whether a sample is random and thus representative of a population, explaining that random sampling tends to produce representative samples and support valid inferences.

c. Determine whether conclusions and generalizations can be made about a population based on a sample.

d. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest, generating multiple samples to gauge variation and making predictions or conclusions about the population.

e. Informally explain situations in which statistical bias may exist.

[MA2019] REG-7 (7) 12 : 12. Make informal comparative inferences about two populations using measures of center and variability and/or mean absolute deviation in context.

Statistics and sampling are important for human performance experiments. Students will learn several sampling types including census, random, stratified random, and convenience. Examples of real-life sampling and experimental design are also shown.

*Note: This video is available in both English and Spanish audio, along with corresponding closed captions.*

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 47 :

[MA2015] AL2 (9-12) 40 :

[MA2015] AL2 (9-12) 42 :

[MA2015] AL2 (9-12) 43 :

[MA2015] PRE (9-12) 45 :45 ) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2]

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'

[MA2015] PRE (9-12) 46 :46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2015] PRE (9-12) 48 :

[MA2015] PRE (9-12) 49 :

[MA2015] ALT (9-12) 44 :

[MA2015] ALT (9-12) 46 :

[MA2015] ALT (9-12) 47 :

47 ) 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-CP2]

[MA2015] AL2 (9-12) 40 :

40 ) 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-CP3]

[MA2015] AL2 (9-12) 42 :

42 ) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [S-CP5]

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.

[MA2015] AL2 (9-12) 43 :

43 ) 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-CP6]

[MA2015] PRE (9-12) 45 :

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 48 :

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

[MA2015] PRE (9-12) 49 :

49 ) Evaluate reports based on data. [S-IC6]

[MA2015] ALT (9-12) 44 :

44 ) 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-CP3]

[MA2015] ALT (9-12) 46 :

46 ) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [S-CP5]

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.

[MA2015] ALT (9-12) 47 :

47 ) 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-CP6]

Learn about the origins and meaning of “p-value,” a statistical measure of the probability that has become a benchmark for success in experimental science, in this video from NOVA: *Prediction by the Numbers*. In the 1920s and 1930s, British scientist Ronald A. Fisher laid out guidelines for designing experiments using statistics and probability to judge results. He proposed that if experimental results were due to chance alone, they would occur less than 5 percent (0.05) of the time. The lower the p-value, the less likely the experimental results were caused by chance. Use this resource to stimulate thinking and questions about the use of statistics and probability to test hypotheses and evaluate experimental results.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 42 : 42 ) 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-ID2]

[MA2015] AL1 (9-12) 43 :43 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [S-ID3]

[MA2015] PRE (9-12) 45 :45 ) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2]

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'

[MA2015] PRE (9-12) 46 :46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2015] PRE (9-12) 49 :

[MA2015] PRE (9-12) 40 :40 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (*Identify unifrom, skewed, and normal distridutions in a set of data. Determine the quartiles and interquartile range for a set of data.*) [S-ID3] (Alabama)

[MA2015] PRE (9-12) 39 :39 ) 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. (*Focus on increasing rigor using standard deviation*). [S-ID2] (Alabama)

[MA2015] AL1 (9-12) 43 :

[MA2015] PRE (9-12) 45 :

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 49 :

49 ) Evaluate reports based on data. [S-IC6]

[MA2015] PRE (9-12) 40 :

[MA2015] PRE (9-12) 39 :

Real-world examples demonstrate the benefits of histograms in this 10-minute video from the Against All Odds statistics series. Data visualization techniques help students understand the practical application of statistics in meteorology and in predicting traffic patterns. Hosted by Pardis Sabeti, this series walks students through understanding how statistics are used in everyday life.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 42 : 42 ) 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-ID2]

[MA2015] AL1 (9-12) 43 :43 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [S-ID3]

[MA2015] PRE (9-12) 44 :

[MA2015] PRE (9-12) 46 :46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2015] PRE (9-12) 48 :48 ) Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. [S-IC5]

[MA2015] PRE (9-12) 40 :40 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (*Identify unifrom, skewed, and normal distridutions in a set of data. Determine the quartiles and interquartile range for a set of data.*) [S-ID3] (Alabama)

[MA2015] PRE (9-12) 39 :39 ) 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. (*Focus on increasing rigor using standard deviation*). [S-ID2] (Alabama)

[MA2015] AL1 (9-12) 43 :

[MA2015] PRE (9-12) 44 :

44 ) Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [S-IC1]

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 48 :

[MA2015] PRE (9-12) 40 :

[MA2015] PRE (9-12) 39 :

Examine a mathematical theory known as the “wisdom of crowds,” which holds that a crowd’s predictive ability is greater than that of an individual, in this video from NOVA: *Prediction by the Numbers*. Sir Francis Galton documented this phenomenon after witnessing a weight-guessing contest more than a hundred years ago at a fair. Statistician Talithia Williams tests Galton’s theory with modern-day fairgoers, asking them to guess the number of jelly beans in a jar. Use this resource to stimulate thinking and questions about the use of statistics in everyday life and to make evidence-based claims about predictive ability.

View Standards
**Standard(s): **
[MA2015] PRE (9-12) 44 :

[MA2015] PRE (9-12) 46 :46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2015] PRE (9-12) 47 :

[MA2015] PRE (9-12) 49 :

44 ) Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [S-IC1]

[MA2015] PRE (9-12) 46 :

[MA2015] PRE (9-12) 47 :

47 ) 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-IC4]

[MA2015] PRE (9-12) 49 :

49 ) Evaluate reports based on data. [S-IC6]

This topic introduces different types of statistical studies (e.g., observational studies, surveys, and experiments) (S-IC.B.3). The role of randomization (i.e., random selection in observational studies and surveys and random assignment in experiments) is addressed. A discussion of random selection (i.e., selecting a sample at random from a population of interest) shows students how selecting participants at random provides a representative sample, thereby allowing conclusions to be generalized from the sample to the population. A discussion of random assignment in experiments, which involves assigning subjects to experimental groups at random, helps students see that random assignment is designed to create comparable groups making it possible to assess the effects of an explanatory variable on a response.

The distinction between population characteristics and sample statistics (first made in Grade 7) is revisited. Scenarios are introduced in which students are asked a statistical question that involves estimating a population mean or a population proportion. For example, students are asked to define an appropriate population, population characteristic, sample, and sample statistics that might be used in a study of the time it takes students to run a quarter mile or a study of the proportion of national parks that contain bald eagle nests.

In this topic, students use data from a random sample to estimate a population mean or a population proportion. Building on what they learned about sampling variability in Grade 7, students use simulation to create an understanding of the margin of error. In Grade 7, students learned that the proportion of successes in a random sample from a population varies from sample to sample due to the random selection process. They understand that the value of the sample proportion is not exactly equal to the value of the population proportion. In Algebra II, they use margin of error to describe how different the value of the sample proportion might be from the value of the population proportion. Students begin by using a physical simulation process to carry out a simulation. Starting with a population that contains successes (using a bag with black beans and white beans), they select random samples from the population and calculate the sample proportion. By doing this many times, they are able to get a sense of what kind of differences are likely. Their understanding should then extend to include the concept of margin of error. Students then proceed to use technology to carry out a simulation. Once students understand the concept of margin of error, they go on to learn how to calculate and interpret it in context (S-IC.A.1, S-IC.B.4). Students also evaluate reports from the media in which sample data are used to estimate a population mean or proportion (S-IC.B.6).

**Note: Although this module is identified as Algebra II in the EngageNY curriculum, it corresponds to the Precalculus Alabama Course of Study.**

View Standards
**Standard(s): **
[MA2015] PRE (9-12) 46 : 46 ) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2015] PRE (9-12) 48 :48 ) Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. [S-IC5]

[MA2015] PRE (9-12) 49 :

[MA2015] PRE (9-12) 48 :

[MA2015] PRE (9-12) 49 :

49 ) Evaluate reports based on data. [S-IC6]

This topic focuses on drawing conclusions based on data from a statistical experiment. Experiments are introduced as investigations designed to compare the effect of two treatments on a response variable. Students revisit the distinction between random selection and random assignment.

When comparing two treatments using data from a statistical experiment, it is important to assess whether the observed difference in group means indicates a real difference between the treatments in the experiment or whether it is possible that there is no difference and that the observed difference is just a by-product of the random assignment of subjects to treatments (S-IC.B.5). To help students understand how this distinction is made, lessons in this topic use simulation to create a randomization distribution as a way of exploring the types of differences they might expect to see by chance when there is no real difference between groups. By understanding these differences, students are able to determine whether an observed difference in means is significant (S-IC.B.5).

Students also critique and evaluate published reports based on statistical experiments that compare two treatments (S-IC.B.6). For example, students read a short summary of an article in the online New England Journal of Medicine describing an experiment to determine if wearing a brace helps adolescents with scoliosis. Then, they watch an online video report for the Wall Street Journal titled “BMW Drivers Really Are Jerks” that describes a study of the relationship between driving behavior and the type of car driven.

**Note: Although this module is identified as Algebra II in the EngageNY curriculum, it corresponds to the Precalculus Alabama Course of Study.**