Chapter 13 finding relationships
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This document contains information about statistics concepts like dependent and independent variables, scatter diagrams, correlation, and regression. It discusses how to interpret a scatter diagram and calculate the correlation coefficient to determine the strength of the linear relationship between two variables. An example looks at the relationship between marks achieved and time studied for 5 students. It finds a strong positive correlation and uses regression to determine the line of best fit and coefficient of determination. The document explains how to use the line of best fit to make predictions, like predicting a student's mark if they studied for a certain number of hours.
Chapter 13 finding relationships
- 1. 2013/05/23 1 STATISTICS 2. X-Kit Textbook Chapter 13 3. Precalculus Textbook Appendix B: Concepts in Statistics Par B.3 Information23 May 2013 • Find information on Edulink on: • Calculation of Semester Marks. • Consultation Time during the June Exam. • Exam Info. • Report any problems with marks to Ms Durandt as soon as possible, CR523. • Semester Tests available at the Collection Facility at the Mathematics Department. • Find all memos on Edulink. • Second Semester? CONTENT Dependent & Independent Variables Scatter Diagrams Correlation Regression TABLE: MARKS ACHIEVEDVERSUSTIME STUDIED STUDENT TIME STUDIED (X) IN HOURS MARKS ACHIEVED (Y) IN % A 2 60 B 5 85 C 1 30 D 4 70 E 2 40 SCATTERDIAGRAM 2, 60 5, 85 1, 30 4, 70 2, 40 0 10 20 30 40 50 60 70 80 90 0 1 2 3 4 5 6 MarksAchieved(%) Time Studied (Hours) Scatter Diagram of marks achieved versus time studied Y-Values INTERPRETINGA SCATTERDIAGRAM 0 10 20 30 40 50 60 70 80 90 0 1 2 3 4 5 6 MarksAchieved(%) Time Studied (Hours) Y-Values Linear (Y-Values)
- 2. 2013/05/23 2 IS THERELATIONSHIPBETWEENVARIABLES STRONGOR WEAK? POSITIVE CORRELATION 0 5 10 15 20 25 30 35 40 0 2 4 6 Y-Values NEGATIVE CORRELATION 0 5 10 15 20 25 0 2 4 6 Y-Values IS THERELATIONSHIPBETWEENVARIABLES STRONGOR WEAK? NO CORRELATION 0 5 10 15 20 25 30 35 0 10 20 30 40 Y-Values NON-LINEAR CORRELATION 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 Y-Values SUMMINGUP CORRELATIONIN A NUMBER • The correlation coefficient 𝑟 is a number that tells us exactly how strong or weak the correlation between two variables is. • Calculate 𝑟 by using the formula: 𝒓 = 𝒙𝒚 − 𝒏𝒙 𝒚 𝒙 𝟐 − 𝒏𝒙 𝟐 𝒚 𝟐 − 𝒏𝒚 𝟐 • Calculate 𝑟 by using your calculator. THE MEANINGOF 𝒓 Perfect Strong Mode- rate Weak No Linear Correlation Weak Mode- rate Strong Perfect -1.00 -0.75 -0.50 -0.25 0.00 +0.25 +0.50 +0.75 +1.00 NEGATIVE CORRELATION POSITIVE CORRELATION CALCULATETHE CORRELATIONCOEFFICIENT: MARKS ACHIEVEDVERSUSTIME STUDIED STUDENT TIME STUDIED (X) IN HOURS MARKS ACHIEVED (Y) IN % A 2 60 B 5 85 C 1 30 D 4 70 E 2 40 SOLUTION 𝒓 = 𝟎. 𝟗𝟑𝟕 Very Strong Positive Linear Relationship
- 3. 2013/05/23 3 MAKE PREDICTIONS Follow the plan: •Find the LINE OF BEST FIT. •Decide how “well” it fits. •From a good fit we can make predictions. LINE OF BESTFIT (REGRESSIONLINE) • Formula 𝒚 = 𝒂 + 𝒃𝒙 • Formula for the regression coefficients: 𝒃 = 𝒙𝒚 − 𝒏𝒙 𝒚 𝒙 𝟐 − 𝒏𝒙 𝟐 𝒂 = 𝒚 − 𝒃𝒙 • Use your calculator to calculate the regression coefficients. CALCULATETHE REGRESSIONCOEFFICIENT: MARKS ACHIEVEDVERSUSTIME STUDIED STUDENT TIME STUDIED (X) IN HOURS MARKS ACHIEVED (Y) IN % A 2 60 B 5 85 C 1 30 D 4 70 E 2 40 LINE OF BESTFIT 𝒚 = 𝟐𝟏. 𝟓 + 𝟏𝟐. 𝟕𝒙 INTERPRETINGA SCATTERDIAGRAM y = 12.685x + 21.481 R² = 0.8777 0 10 20 30 40 50 60 70 80 90 0 1 2 3 4 5 6 MarksAchieved(%) Time Studied (Hours) Y-Values Y-Values Linear (Y-Values) MEASURINGHOW WELL THELINE FITS • How well does our line fit the real data? How accurate is our model? • 𝑟 , the CORRELATION COEFFICIENT tells us how STRONG the relationship is between two variables, or how closely the data fits our line. • 𝑟2 , the COEFFICIENT OF DETERMINATION measure the ACCURACY of our predictions. For a perfect fit 𝑟2 = 1 , closer to zero indicate a poorer fit. • The coefficient of determination tells us that 87.8% of the variation in students’ marks is linked to the amount of time they spend studying. The other 12.2% is due to other factors, like intelligence levels.
- 4. 2013/05/23 4 PREDICTINGFROMTHELINE OF “BEST FIT” If your friend only study for 2.5 hours, will he pass the test? 𝒚 = 𝟐𝟏. 𝟓 + 𝟏𝟐. 𝟕𝒙 HOMEWORK •Example X-Kit textbook page 310 – 311. •Practise for your exams page 312 number 1, 2, 3, 4, 5, & 6. •Par B.3 (page B14) all odd number questions.