Using A Calculator To Investigate Whether A Linear, Quadratic ...

investigate whether a linear, quadratic or exponential function best fits a set of bivariate numerical data Jackie Scheiber RADMASTE Wits University [email protected] 1

Curriculum References GRADE 9 Draw a variety of graphs by hand/technology to display and interpret data including: Bar graphs and double bar graphs Histograms with given and own intervals Pie charts Broken-line graphs

Scatter plots o the scatter plot allows one to see trends and make predictions, as well as identify outliers in the data 2 GRADE 12 a) Represent bivariate data as a scatter plot and suggest intuitively and by simple investigation whether a linear,

quadratic or exponential function would best fit the data b) Use a calculator to calculate the linear regression line which best fits a given set of bivariate data c) Use a calculator to calculate the correlation coefficient of a set of bivariate numerical data and make relevant deductions. 3

Bivariate Data BIVARIATE DATA each item in the population has TWO measurements associated with it We can plot bivariate data on a SCATTER PLOT (or scatter diagram or scatter graph or scatter chart) The scatter graph shows whether there is an association or CORRELATION between the two variables 4

Some Types of Correlation Positive Linear Correlation Negative Linear Correlation No Correlation

Non-linear Correlation 5 TASK 1 6 1) 7

2) Negative correlation As the date increases, the time taken decreases 8 3 a) A = 23,746 23,75 B = - 0,0069 - 0,007

Equation of the linear regression line is y = 23,75 0,007 x 9 3 b) 10 3 c)

11 4 a) A = - 418,943 - 418,94 B = 0,439 0,44 C = - 0,00011 - 0,001 Equation of the quadratic regression function: y = - 418,94 + 0,44 x 0,0001 x2 12

4b) Year 1960 1968 1988 1999 2005 2008 Time 10,0 10,0 9,90 9,80 9,73 9,70 3 1 13 4 c)

14 5 a) A = 40,294 40,29 B = 0,9992 0,999 Equation of the exponential regression function: y = 40,29 . (0,999)x 15

5 b) Year 1960 1968 1988 1999 2005 2008 Time 10,0 10,0 9,86 9,79 9,74 9,72 6 0 16 5 c) 17

6) Year 1960 1968 1988 1999 2005 2008

Actual time (s) 10,0 9,95 9,92 9,79 9,77 9,69

Value of the regression function Linear Quadratic Exponentia l 10,06 10,00 9,86

9,79 9,74 9,72 10,03 10,01 9,90 9,80 9,73 9,70

10,06 10,00 9,86 9,79 9,74 9,72 18 7) 19

7) continued Compare the Linear and Exponential regression functions similar results rather use the straight line as it is simpler Compare the Linear and Quadratic regression functions similar results but part of the data may appear quadratic, but the entire set may be less symmetric. The Linear regression function seems to suit the given data best.

20 TASK 2 21 1) 22

2) a) y = 69,14 + 0,06 x b) y = 75,39 . (1,0004)x 23 3) 24 4)

Distanc e Measur ed speed Linear model Exponenti

al model 1 000 m ~ 145 m/s 112,4 m/s 2 700 m ~ 233 m/s

3 700 m ~ 274 m/s 127,55 m/ s 226,85 m/ s 285,26 m/ s

220,25 m/s 327,62 m/s 25 5) As can be seen in the table, each time the values from the linear model are closest to the measured values so the linear model fits the data better.

26

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