Introduction The fit of a linear function to a set of data can be assessed by analyzing residuals. A residual is the vertical distance between an observed.

Slides:



Advertisements
Similar presentations
5.4 Correlation and Best-Fitting Lines
Advertisements

Introduction The fit of a linear function to a set of data can be assessed by analyzing residuals. A residual is the vertical distance between an observed.
Introduction When linear functions are used to model real-world relationships, the slope and y-intercept of the linear function can be interpreted in context.
~adapted from Walch Education A scatter plot that can be estimated with a linear function will look approximately like a line. A line through two points.
1.5 Scatter Plots and Least-Squares Lines
Introduction Tables and graphs can be represented by equations. Data represented in a table can either be analyzed as a pattern, like the data presented.
Residuals and Residual Plots Most likely a linear regression will not fit the data perfectly. The residual (e) for each data point is the ________________________.
Biostatistics Unit 9 – Regression and Correlation.
Line of Best Fit 4.2 A. Goal Understand a scatter plot, and what makes a line a good fit to data.
Correlation The apparent relation between two variables.
Solving Problems Given Functions Fitted to Data/Interpreting Slope and y-intercept Key Terms: Linear Fit Slope Scatter Plot 1 4.6: Solving Problems Given.
7-3 Line of Best Fit Objectives
1.5 Scatter Plots and Least-Squares Lines Objectives : Create a scatter plot and draw an informal inference about any correlation between the inference.
Lesson 4.6 Best Fit Line Concept: Using & Interpreting Best Fit Lines EQs: - How do we determine a line of best fit from a scatter plot? (S.ID.6 a,c) -What.
Check it out! : Fitting Linear Functions to Data.
Solve the equation for y. SOLUTION EXAMPLE 2 Graph an equation Graph the equation –2x + y = –3. –2x + y = –3 y = 2x –3 STEP 1.
Linear Best Fit Models Learn to identify patterns in scatter plots, and informally fit and use a linear model to solve problems and make predictions as.
REGRESSION MODELS OF BEST FIT Assess the fit of a function model for bivariate (2 variables) data by plotting and analyzing residuals.
4.4 – SCATTER PLOTS AND LINES OF FIT Today’s learning goal is that students will be able to: interpret scatter plots, identify correlations between data.
4.2.4: Warm-up, Pg.81 The data table to the right shows temperatures in degrees Fahrenheit taken at 7:00 A. M. and noon on 8 different days throughout.
Unit 4 Part B Concept: Best fit Line EQ: How do we create a line of best fit to represent data? Vocabulary: R – correlation coefficient y = mx + b slope.
4.3.2: Warm-up, P.107 A new social networking company launched a TV commercial. The company tracked the number of users in thousands who joined the network.
Warm up 1. Calculate the slope of a line passing through (2, 3) and (1, 5). 2. Write the equation of a line given a slope of 3 and a y-intercept of 4 (hint:
Scatter Plots and Equations of Lines Chapter 6 Section 7.
Introduction The relationship between two variables can be estimated using a function. The equation can be used to estimate values that are not in the.
Scatter Plots and Lines of Fit
Lesson 4.5 Topic/ Objective: To use residuals to determine how well lines of fit model data. To use linear regression to find lines of best fit. To distinguish.
Day 38 – Line of Best Fit (Day 2)
distance prediction observed y value predicted value zero
Residuals Algebra.
Lines of Best Fit When data show a correlation, you can estimate and draw a line of best fit that approximates a trend for a set of data and use it to.
Population (millions)
SIMPLE LINEAR REGRESSION MODEL
Residuals From the Carnegie Foundation math.mtsac.edu/statway/lesson_3.3.1_version1.5A.
Unit 4 EOCT Review.
Lesson 4.8 – Interpreting the Correlation Coefficient and Distinguishing between Correlation & Causation EQ: How do you calculate the correlation coefficient?
Introduction The fit of a linear function to a set of data can be assessed by analyzing residuals. A residual is the vertical distance between an observed.
Homework: Residuals Worksheet
Math 8C Unit 3 – Statistics
Suppose the maximum number of hours of study among students in your sample is 6. If you used the equation to predict the test score of a student who studied.
Splash Screen.
2.5 Correlation and Best-Fitting Lines
2.6 Draw Scatter Plots and Best-Fitting Lines
Residuals Learning Target:
Introduction Real-world contexts that have two variables can be represented in a table or graphed on a coordinate plane. There are many characteristics.
Lesson Objective: I will be able to …
Correlation describes the type of relationship between two data sets.
Residuals and Residual Plots
~adapted for Walch Education
Day 37 Beginner line of the best fit
Functions and Their Graphs
Prediction and Accuracy
Scatter Plots and Equations of Lines
High School – Pre-Algebra - Unit 8
Review Homework.
Day 38 – Line of Best Fit (Day 2)
Correlation describes the type of relationship between two data sets.
Residuals From the Carnegie Foundation math.mtsac.edu/statway/lesson_3.3.1_version1.5A.
A. Draw a trend line. It will be easier to write an equation
Correlation describes the type of relationship between two data sets.
Lesson 2.2 Linear Regression.
Which graph best describes your excitement for …..
9/27/ A Least-Squares Regression.
Applying linear and median regression
Draw Scatter Plots and Best-Fitting Lines
Residuals From the Carnegie Foundation math.mtsac.edu/statway/lesson_3.3.1_version1.5A.
Correlation describes the type of relationship between two data sets.
Residuals and Residual Plots
Graphing Linear Equations
Presentation transcript:

Introduction The fit of a linear function to a set of data can be assessed by analyzing residuals. A residual is the vertical distance between an observed data value and an estimated data value on a line of best fit. Representing residuals on a residual plot provides a visual representation of the residuals for a set of data. A residual plot contains the points: (x, residual for x). A random residual plot, with both positive and negative residual values, indicates that the line is a good fit for the data. If the residual plot follows a pattern, such as a U-shape, the line is likely not a good fit for the data. 4.2.3: Analyzing Residuals

Key Concepts A residual is the distance between an observed data point and an estimated data value on a line of best fit. For the observed data point (x, y) and the estimated data value on a line of best fit (x, y0), the residual is y – y0. A residual plot is a plot of each x-value and its corresponding residual. For the observed data point (x, y) and the estimated data value on a line of best fit (x, y0), the point on a residual plot is (x, y – y0). 4.2.3: Analyzing Residuals

Key Concepts, continued A residual plot with a random pattern indicates that the line of best fit is a good approximation for the data. A residual plot with a U-shape indicates that the line of best fit is not a good approximation for the data. 4.2.3: Analyzing Residuals

Common Errors/Misconceptions incorrectly finding the residual incorrectly plotting points on the residual plot 4.2.3: Analyzing Residuals

Guided Practice Example 1 Pablo’s science class is growing plants. He recorded the height of his plant each day for 10 days. The plant’s height, in centimeters, over that time is listed in the table to the right. Day Height in centimeters 1 3 2 5.1 7.2 4 8.8 5 10.5 6 12.5 7 14 8 15.9 9 17.3 10 18.9 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Pablo determines that the function y = 1.73x + 1.87 is a good fit for the data. How close is his estimate to the actual data? Approximately how much does the plant grow each day? 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Create a scatter plot of the data. Let the x-axis represent days and the y-axis represent height in centimeters. Height Days 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Draw the line of best fit through two of the data points. A good line of best fit will have some points below the line and some above the line. Use the graph to initially determine if the function is a good fit for the data. 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Height Days 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Find the residuals for each data point. The residual for each data point is the difference between the observed value and the estimated value using a line of best fit. Evaluate the equation of the line at each value of x. 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued y = 1.73x + 1.87 1 y = 1.73(1) + 1.87 = 3.6 2 y = 1.73(2) + 1.87 = 5.33 3 y = 1.73(3) + 1.87 = 7.06 4 y = 1.73(4) + 1.87 = 8.79 5 y = 1.73(5) + 1.87 = 10.52 6 y = 1.73(6) + 1.87 = 12.25 7 y = 1.73(7) + 1.87 =13.98 8 y = 1.73(8) + 1.87 = 15.71 9 y = 1.73(9) + 1.87 = 17.44 10 y = 1.73(10) + 1.87 = 19.17 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Next, find the difference between each observed value and each calculated value for each value of x. x Residual 1 3 – 3.6 = –0.6 2 5.1 – 5.33 = –0.23 3 7.2 – 7.06 = 0.14 4 8.8 – 8.79 = 0.01 5 10.5 – 10.52 = –0.02 6 12.5 – 12.25 = 0.25 7 14 – 13.98 = 0.02 8 15.9 – 15.71 = 0.19 9 17.3 – 17.44 = –0.14 10 18.9 – 19.17 = –0.27 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Plot the residuals on a residual plot. Plot the points (x, residual for x). 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued Describe the fit of the line based on the shape of the residual plot. The plot of the residuals appears to be random, with some negative and some positive values. This indicates that the line is a good line of fit. 4.2.3: Analyzing Residuals

✔ Guided Practice: Example 1, continued Use the equation to estimate the centimeters grown each day. The change in the height per day is the centimeters grown each day. In the equation of the line, the slope is the change in height per day. The plant is growing approximately 1.73 centimeters each day. ✔ 4.2.3: Analyzing Residuals

Guided Practice: Example 1, continued http://walch.com/ei/CAU4L2S3GraphResid 4.2.3: Analyzing Residuals

Guided Practice Example 2 Lindsay created the table to the right showing the population of fruit flies over the last 10 weeks. Week Number of flies 1 50 2 78 3 98 4 122 5 153 6 191 7 238 8 298 9 373 10 466 4.2.3: Analyzing Residuals

Guided Practice: Example 2, continued She estimates that the population of fruit flies can be represented by the equation y = 46x – 40. Using residuals, determine if her representation is a good estimate. 4.2.3: Analyzing Residuals

Guided Practice: Example 2, continued Find the estimated population at each x-value. Evaluate the equation at each value of x. x y = 46x – 40 1 46(1) – 40 = 6 2 46(2) – 40 = 52 3 46(3) – 40 = 98 4 46(4) – 40 = 144 5 46(5) – 40 = 190 6 46(6) – 40 = 236 7 46(7) – 40 = 282 8 46(8) – 40 = 328 9 46(9) – 40 = 374 10 46(10) – 40 = 420 4.2.3: Analyzing Residuals

Guided Practice: Example 2, continued Find the residuals by finding each difference between the observed population and estimated population. x Residual 1 50 – 6 = 44 2 78 – 52 = 26 3 98 – 98 = 0 4 122 – 144 = –22 5 153 – 190 = –37 6 191 – 236 = –45 7 238 – 282 = –44 8 298 – 328 = –30 9 373 – 374 = –1 10 466 – 420 = 46 4.2.3: Analyzing Residuals

Guided Practice: Example 2, continued Create a residual plot. Plot the points (x, residual for x). 4.2.3: Analyzing Residuals

✔ Guided Practice: Example 2, continued Analyze the residual plot to determine if the equation is a good estimate for the population. The residual plot has a U-shape. This indicates that a non-linear estimation would be a better fit for this data set. The shape of the residual plot indicates that the equation y = 46x – 40 is not a good estimate for this data set. ✔ 4.2.3: Analyzing Residuals

Guided Practice: Example 2, continued http://walch.com/ei/CAU4L2S3Resid 4.2.3: Analyzing Residuals