Using a calculator to investigate whether a linear, quadratic or exponential function best fits a set of bivariate numerical data Jackie Scheiber RADMASTE Wits University 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

3 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.

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 CorrelationNegative Linear Correlation No CorrelationNon-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 x 2 12

4b) Year Time10,0310,019,909,809,739,70 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 Time10,0610,009,869,799,749,72 16

5 c) 17

6) Year Actual time (s) Value of the regression function LinearQuadraticExponential ,010,0610,0310, ,9510,0010,0110, ,929,869,909, ,79 9,809, ,779,749,739, ,699,729,709,72 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) Distance Measured speed Linear model Exponential model m~ 145 m/s127,55 m/s112,4 m/s m~ 233 m/s226,85 m/s220,25 m/s m~ 274 m/s285,26 m/s327,62 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