10.1: Scatter Plots & Trend Lines. Essential Question: How can you describe the relationship between two variables and use it to make predictions?

Transcription

1 10.1: Scatter Plots & Trend Lines Essential Question: How can you describe the relationship between two variables and use it to make predictions?

2 Vocab Two-variable data: two data points, one individual/object. For example: weight and height of a 9 th grader or MYA score and Math % grade of an Algebra 1 student. Scatterplot: Visual representation of twovariable data. One value is plotted on the x-axis and the other on the y-axis.

3 Vocab Correlation: the mathematical strength of a relationship between two-variable data. Correlation measures the strength and direction. Positive Correlation: both variables increase together. Negative Correlation: one variable increases while the other decrease. No correlation: no relationship or pattern in scatterplot.

4

5

6 What is Correlation? Correlation values are always between -1 and 1. Correlation is abbreviated: r The strength of the linear relationship increases as r moves away from 0 towards -1 or 1. r = 0 ; no correlation r = 1; positive, very strong correlation r = -1; negative, very strong correlation

7 Which of the following usually has a positive correlation? Select all that apply.

8

9

10 What does r mean? R Value Strength -1 Perfectly linear; negative Strong negative relationship Moderately strong negative relationship Weak negative relationship 0 nonexistent 0.25 Weak positive relationship 0.50 Moderately strong positive relationship 0.75 Strong positive relationship 1 Perfectly linear; positive

11 Estimate r. Describe the correlation.

12 Estimate r. Describe the correlation.

13 Estimate r. Describe the correlation.

14

15

16 Vocab Line of best fit: a line through a set of twovariable data that illustrates the correlation. We will use a visual method. Using two points on your line, you can write the linear equation for the line of best fit.

17 Create a Line of Best Fit 1. Flip back to your Scatterplot about city temperature and latitude. 2. Draw a line that best fits the data. It does not need to go through any data points, but can. 3. Identify two points on the line and calculate the slope. 4. Use the slope-intercept form, (x, y) coordinate and the slope to write the equation.

18 Plot the points and calculate the line of best fit.

19

20

21

22

23 Making Predictions using Linear Regression We can predict values using linear regression. We can plug in domain values to determine predicted outputs/range values. Interpolation- predictions involving values between the minimum and maximum of the domain Extrapolation- predictions involving values outside the minimum or maximum. Use caution!!!!

24 Making Predictions What is the boiling point of water in Mexico City? The altitude is 7943 ft. What is the boiling point of water in Fargo, North Dakota? The altitude is 3000 ft. Which prediction is more reliable? Why?

25

26 Vocab: Correlation vs. Causation Correlation- two variables that are mathematically related. Examples: Causation- two variables where one is CAUSED by the other. Need a scientific experiment to prove causation. Examples :

27 Correlation or Causation? 1. As outdoor temperatures rise, so do ice cream sales. 2. As shoe size increases, so do reading abilities. 3. As traffic on Biscayne Blvd increases, so do ATM tardies. 4. As a person s height increases, so does his/her weight. 5. As a student s test scores increase, so does his/her grade.

28 Likely, Doubtful or Unclear A traffic official in a major metropolitan area notice that the more profitable toll bridges into the city are those with the slowest average crossing speeds. The variable are and. It is that increased profit causes slower crossing speed. It is that slower crossing speed causes an increase in profits.

29

30

31 10.2: Fitting a Linear Model to Data Essential Question: How can you use the linear regression function and residuals to evaluate the accuracy for a line of best fit?

32 Residuals: difference between an observed value of the response variable and the value predicted by the regression line. Tells how accurate or not a line of best fit is. Formula: Residual: Actual- Predicted Residuals

33 How to Calculate the Residual 1. Calculate the predicted value, by plugging in x to the best fit line. 2. Determine the observed/actual value. 3. Subtract.

34 Create a Scatterplot A, age in years H, height in inches

35 Calculate the Residuals Using the line H = 3.5A + 23, calculate the residuals. A, age in years H, height in inches Predicted Values Residuals

36

37 Residual Plots A residual plot is a scatterplot of the residuals against the explanatory variable. Residual plots help us assess how well a regression line fits the data.

38 Residual Plots: Good Fit

39 Residual Plots: Bad Fit

40 Plot the Residuals

41 Evaluate the Residual Plot 1. Use your residual plot to discuss how well the student s line fits the data. 2. Use the student s line to predict the height of a 20- year-old man. Discuss the reasonableness of the result.

42