Lesson 18 | Descriptive Statistics | 9th Grade Mathematics

Objective

Use residuals to assess the strength of the model for a data set.

Common Core Standards

Core Standards

The core standards covered in this lesson

HSS-ID.B.6b — Informally assess the fit of a function by plotting and analyzing residuals.

Interpreting Categorical and Quantitative Data

HSS-ID.B.6b — Informally assess the fit of a function by plotting and analyzing residuals.
HSS-ID.B.6c — Fit a linear function for a scatter plot that suggests a linear association.

Interpreting Categorical and Quantitative Data

HSS-ID.B.6c — Fit a linear function for a scatter plot that suggests a linear association.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Explain that the least squares regression line is the predicted model for the data, whereas the data points are the actual data.
Describe that while you can fit various functions to a set of data and develop an algebraic model, the focus in this unit is on fitting and analyzing a linear model.
Relate the standard deviation (distance from the mean) with comparing the actual data points in a scatterplot from the distance from the line of best fit.
Calculate the residuals between the actual data and the line of best fit.
Describe that by plotting and analyzing the residuals, you can make a conjecture about whether a particular function is appropriate for a set of data and variables.
Analyze a residual plot for randomness. If a residual plot has a random pattern, the function is likely an appropriate model for the relationship between the two variables.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Below is a set of data.
On the left is an incorrect line of best fit, and on the right is the least squares regression line.

$${y=x+1}$$$${y=.9x+2.5}$$
A student explains that the line on the left is more appropriate, because it goes through three points in the data set.

What calculations can you do to convince this student that even though the line of best fit does not go through any of the points on the graph on the right, it is a better model for the data?

Guiding Questions

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Problem 2

What function will be the best fit for this set of data? How can we use residuals to determine this?

Guiding Questions

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Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Meerkats have a gestation time of days.

Use the equation of the least squares line, $${y=6.643+0.03974x}$$ , to predict the longevity of the meerkat. Remember $$x$$ equals the gestation time in days and $$y$$ equals the longevity in years.
Approximately how close might your prediction be to the actual longevity of the meerkat? What was it (from class) that told you roughly how close a prediction might be to the true value?
The longevity of the meerkat is actually 10 years. Use this value and the predicted value that you calculated in part a. to find the residual for the meerkat.

References

EngageNY Mathematics Algebra I > Module 2 > Topic D > Lesson 15 — Exit Ticket #1

Algebra I > Module 2 > Topic D > Lesson 15 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

EngageNY Mathematics Algebra I > Module 2 > Topic D > Lesson 15 — Problem Set
Mathematics Vision Project: Secondary Mathematics One Module 9: Modeling Data — Lesson 8: Rocking the Residuals
MARS Formative Assessment Lessons for High School Devising a Measure: Correlation
Illustrative Mathematics Restaurant Bill and Party Size
New Visions for Public Schools Connecting Representations - Residuals

Lesson 17

Lesson 19

Lesson Map

Topic A: Descriptive Statistics in Univariate Data

Describe statistics. Represent data in frequency graphs and identify the center of a data set.

HSS-IC.A.1 HSS-ID.A.1 HSS-ID.A.2

Describe center and spread. Represent data in a box plot (box-and-whisker plot) and calculate the center and spread.

HSS-ID.A.1 HSS-ID.A.2

Represent data in a histogram and calculate the center. Identify when the median and mean are not the same value.

HSS-ID.A.1

Describe the shape of the data in box plots and histograms. Choose an appropriate measure of center (or an appropriate shape) based on the shape and the relationship between the mean and the median.

HSS-ID.A.2 HSS-ID.A.3

Calculate and interpret the spread (variance) of a data set.

HSS-ID.A.3 HSS-ID.A.4

Calculate the standard deviation and compare two symmetrical distributions based on the mean and standard deviation.

HSS-ID.A.2 HSS-ID.A.4

Interpret the standard deviation and interquartile range.

HSS-ID.A.2 HSS-ID.A.4

Calculate population percentages using the standard deviation.

HSS-ID.A.4

Given summary statistics, describe the best measures of center and spread. Describe reasoning.

HSS-ID.A.2

Develop and answer statistical questions through data analysis of existing data using appropriate statistical measures and displays. (Part 1/3)

HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3 HSS-ID.A.4

Develop and answer statistical questions through data analysis of existing data using appropriate statistical measures and displays. (Part 2/3)

HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3 HSS-ID.A.4

Develop and answer statistical questions through data analysis of existing data using appropriate statistical measures and displays. (Part 3/3)

HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3 HSS-ID.A.4

Topic B: Descriptive Statistics in Bivariate Data

Define categorical and numerical data. Create two-way tables to organize bivariate categorical data.

HSS-ID.B.5

Describe relative and relative conditional frequencies of two-way tables.

HSS-ID.B.5

Create scatterplots and identify function shapes in scatterplots.

HSS-ID.B.6

Calculate, with technology, the correlation coefficient for a data set. Explain why correlation does not determine causation.

HSS-ID.C.8 HSS-ID.C.9

Determine the function of best fit and create a linear equation from least squares regression using technology.

HSS-ID.B.6a HSS-ID.B.6b HSS-ID.C.7

Use residuals to assess the strength of the model for a data set.

HSS-ID.B.6b HSS-ID.B.6c

Describe the relationship between two quantitative variables in a contextual situation represented in a scatterplot using the correlation coefficient, least squares regression, and residuals as evidence.

HSS-ID.B.6a HSS-ID.C.7 HSS-ID.C.9

Develop and answer statistical questions through data analysis of existing data using appropriate statistical measures and displays. (Part 1/3)

HSS-ID.B.6 HSS-ID.C.7 HSS-ID.C.8 HSS-ID.C.9

Develop and answer statistical questions through data analysis of existing data using appropriate statistical measures and displays. (Part 2/3)

HSS-ID.B.6 HSS-ID.C.7 HSS-ID.C.8 HSS-ID.C.9 N.Q.A.1

Develop and answer statistical questions through data analysis of existing data using appropriate statistical measures and displays. (Part 3/3)

HSS-ID.B.6 HSS-ID.C.7 HSS-ID.C.8 HSS-ID.C.9

Descriptive Statistics

Objective

Common Core Standards

Core Standards

Interpreting Categorical and Quantitative Data

Interpreting Categorical and Quantitative Data

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem Set

Target Task

References

Additional Practice

Lesson Map

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