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# Chapter 13 finding relationships

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### Chapter 13 finding relationships

1. 1. 2013/05/231STATISTICS2. X-Kit TextbookChapter 133. Precalculus TextbookAppendix B: Concepts in StatisticsPar B.3Information23 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 Durandtas soon as possible, CR523.• Semester Tests available at the Collection Facilityat the Mathematics Department.• Find all memos on Edulink.• Second Semester?CONTENTDependent &IndependentVariablesScatter DiagramsCorrelationRegressionTABLE: MARKS ACHIEVEDVERSUSTIMESTUDIEDSTUDENT TIME STUDIED(X) IN HOURSMARKSACHIEVED (Y)IN %A 2 60B 5 85C 1 30D 4 70E 2 40SCATTERDIAGRAM2, 605, 851, 304, 702, 4001020304050607080900 1 2 3 4 5 6MarksAchieved(%)Time Studied (Hours)Scatter Diagram of marks achieved versus time studiedY-ValuesINTERPRETINGA SCATTERDIAGRAM01020304050607080900 1 2 3 4 5 6MarksAchieved(%)Time Studied (Hours)Y-ValuesLinear (Y-Values)
2. 2. 2013/05/232IS THERELATIONSHIPBETWEENVARIABLESSTRONGOR WEAK?POSITIVE CORRELATION05101520253035400 2 4 6Y-ValuesNEGATIVE CORRELATION05101520250 2 4 6Y-ValuesIS THERELATIONSHIPBETWEENVARIABLESSTRONGOR WEAK?NO CORRELATION051015202530350 10 20 30 40Y-ValuesNON-LINEAR CORRELATION00.511.522.533.50 1 2 3Y-ValuesSUMMINGUP CORRELATIONIN A NUMBER• The correlation coefficient 𝑟 is a number thattells us exactly how strong or weak thecorrelation between two variables is.• Calculate 𝑟 by using the formula:𝒓 =𝒙𝒚 − 𝒏𝒙 𝒚𝒙 𝟐 − 𝒏𝒙 𝟐 𝒚 𝟐 − 𝒏𝒚 𝟐• Calculate 𝑟 by using your calculator.THE MEANINGOF 𝒓Perfect Strong Mode-rateWeak No LinearCorrelationWeak Mode-rateStrong Perfect-1.00 -0.75 -0.50 -0.25 0.00 +0.25 +0.50 +0.75 +1.00NEGATIVE CORRELATION POSITIVE CORRELATIONCALCULATETHE CORRELATIONCOEFFICIENT:MARKS ACHIEVEDVERSUSTIME STUDIEDSTUDENT TIME STUDIED(X) IN HOURSMARKSACHIEVED (Y)IN %A 2 60B 5 85C 1 30D 4 70E 2 40SOLUTION𝒓 = 𝟎. 𝟗𝟑𝟕Very Strong Positive LinearRelationship
3. 3. 2013/05/233MAKE PREDICTIONSFollow 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 𝒚𝒙 𝟐 − 𝒏𝒙 𝟐𝒂 = 𝒚 − 𝒃𝒙• Use your calculator to calculate the regressioncoefficients.CALCULATETHE REGRESSIONCOEFFICIENT:MARKS ACHIEVEDVERSUSTIME STUDIEDSTUDENT TIME STUDIED(X) IN HOURSMARKSACHIEVED (Y)IN %A 2 60B 5 85C 1 30D 4 70E 2 40LINE OF BESTFIT𝒚 = 𝟐𝟏. 𝟓 + 𝟏𝟐. 𝟕𝒙INTERPRETINGA SCATTERDIAGRAMy = 12.685x + 21.481R² = 0.877701020304050607080900 1 2 3 4 5 6MarksAchieved(%)Time Studied (Hours)Y-ValuesY-ValuesLinear (Y-Values)MEASURINGHOW WELL THELINE FITS• How well does our line fit the real data? How accurate isour model?• 𝑟 , the CORRELATION COEFFICIENT tells us how STRONGthe relationship is between two variables, or how closelythe data fits our line.• 𝑟2, the COEFFICIENT OF DETERMINATION measure theACCURACY 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 thevariation in students’ marks is linked to the amount oftime they spend studying. The other 12.2% is due toother factors, like intelligence levels.
4. 4. 2013/05/234PREDICTINGFROMTHELINE OF “BEST FIT”If your friend only study for 2.5 hours,will he pass the test?𝒚 = 𝟐𝟏. 𝟓 + 𝟏𝟐. 𝟕𝒙HOMEWORK•Example X-Kit textbook page 310 – 311.•Practise for your exams page 312 number1, 2, 3, 4, 5, & 6.•Par B.3 (page B14) all odd numberquestions.