# Making Inferences

• Understand that statistical methods are used to draw conclusions from data.
• Understand that the validity of data-based conclusions depends on the quality of the data and how the data were collected.
• Critique and evaluate data-based claims that appear in popular media.
• Distinguish between observational studies, surveys and experiments.
• Explain why random selection is important in the design of observational studies and surveys.
• Explain why random assignment is important in the design of statistical experiments.
• Calculate and interpret the standard deviation as a measure of variability.
• Use the normal distribution as a model for data distributions that are approximately symmetric and bell-shaped.
• Use the least squares regression line to model linear relationships in bivariate numerical data.
• Understand sampling variability in the context of estimating a population or a population mean.
• Use data from a random sample to estimate a population proportion.
• Use data from a random sample to estimate a population mean.
• Calculate and interpret margin of error in context.
• Understand the relationship between sample size and margin of error.
• Given data from a statistical experiment, create a randomization distribution.
• Use a randomization distribution to determine if there is a significant difference between two experimental conditions.

This unit builds on the foundation of One Variable Statistics and Bivariate Statistics, as well as students’ work with statistics in grade 7. In particular, the concepts of sampling variability and distributions introduced in earlier units are critical to understanding the process of drawing conclusions from data, which is central to this unit.

In this unit, students revisit and extend their understanding of variability in data and of ways to describe variability in data. Students calculate and interpret the standard deviation, first introduced at a conceptual level in One Variable Statistics, as a way to quantify variability.

Students use distributions to describe variability. Normal distributions are introduced as a way to model data distributions that are bell-shaped and approximately symmetric. Students calculate and interpret areas under a normal curve in the context of modeling a data distribution.

Students explore the reasoning that allows them to draw conclusions based on data from statistical studies. They learn the distinction between an observational study and a statistical experiment. They then use data from a random sample from a population to estimate the values of population characteristics such as a population mean or a population proportion. Students develop the understanding that such estimates are subject to sampling variability. The notion of margin of error is introduced as a way of quantifying the uncertainty associated with an estimate of a population characteristic.

Students also explore the important idea of “statistical significance” as they use data from statistical experiments to determine if there is a significant difference between experimental conditions.

The standards addressed in this unit (especially S-IC.B.4 and S-IC.B.5) are conceptually complex and will require several weeks to fully develop.

Together with One Variable StatisticsBivariate Statistics, and Probability, this unit provides a capstone experience in statistics for grades 6–12 and a solid foundation for an AP Statistics course or a college level introductory statistics course.

## Sections

A2.7.0 Pre-unit diagnostic assessment

#### Summary

Assess students’ ability to
• construct and interpret a graphical display;
• calculate and interpret a sample proportion;
• calculate and interpret a sample mean;
• fit a line to bivariate data.

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A2.7.1 Drawing reasonable conclusions

#### Summary

• Understand that statistical methods are used to draw conclusions from data.
• Understand that the validity of data-based conclusions depends on the quality of the data and how the data were collected.
• Critique and evaluate data-based claims that appear in popular media.

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A2.7.2 Collecting data and types of statistical studies

#### Summary

• Distinguish between observational studies, surveys and experiments.
• Explain why random selection is important in the design of observational studies and surveys.
• Explain why random assignment is important in the design of statistical experiments.

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A2.7.3 Describing data distributions

#### Summary

• Calculate and interpret the standard deviation as a measure of variability.
• Use the normal distribution as a model for data distributions that are approximately symmetric and bell-shaped.
• Use the least squares regression line to model linear relationships in bivariate numerical data.

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A2.7.4 Mid-unit assessment

#### Summary

Assess students’ ability to
• given a description of a statistical study, identify the study type (observational study, survey, or experiment);
• determine what type of statistical study would produce data that could be used to answer a given question;
• distinguish between data distributions for which it would be reasonable to use the normal distribution as a model and those for which it would not be reasonable;
• find an area under a normal curve and interpret it in the context of modeling a data distribution.

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A2.7.5 Drawing conclusions based on data from a random sample

#### Summary

• Understand sampling variability in the context of estimating a population or a population mean.
• Use data from a random sample to estimate a population proportion.
• Use data from a random sample to estimate a population mean.
• Calculate and interpret margin of error in context.
• Understand the relationship between sample size and margin of error.

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A2.7.6 Drawing conclusions based on data from a statistical experiment

#### Summary

• Given data from a statistical experiment, create a randomization distribution.
• Use a randomization distribution to determine if there is a significant difference between two experimental conditions.

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A2.7.7 Summative assessment

#### Summary

Assess students’ ability to
• distinguish between an observational study and an experiment;
• use an appropriate normal distribution to model a data distribution;
• estimate a population proportion and interpret a margin of error in context;
• given a simulated sampling distribution, estimate a margin of error;
• given data from a statistical experiment, create a randomization distribution and use it to determine if there is a significant difference between experimental conditions.

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