using a calculator to investigate whether a linear, quadratic or exponential function best fits a...

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Using a calculator to investigate whether a linear, quadratic or

exponential function best fits a set of

bivariate numerical data

Jackie ScheiberRADMASTE

Wits UniversityJackie.scheiber@wits.ac.za

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Curriculum References

GRADE 9Draw 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

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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.

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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

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Some Types of Correlation

Positive Linear Correlation

Negative Linear Correlation

No Correlation Non-linear Correlation

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TASK 1

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1)

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2)• Negative correlation• As the date increases, the time taken

decreases

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3 a)A = 23,746 … ≈ 23,75B = - 0,0069… ≈ - 0,007

Equation of the linear regression line is y = 23,75 – 0,007 x

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3 b)

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3 c)

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4 a)A = - 418,943 … ≈ - 418,94B = 0,439 … ≈ 0,44C = - 0,00011 … ≈ - 0,001

Equation of the quadratic regression function: y = - 418,94 + 0,44 x – 0,0001 x2

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4b)Year 1960 1968 1988 1999 2005 2008Time 10,0

310,0

19,90 9,80 9,73 9,70

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4 c)

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5 a)A = 40,294 … ≈ 40,29B = 0,9992… ≈ 0,999

Equation of the exponential regression function: y = 40,29 . (0,999)x

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5 b)Year 1960 1968 1988 1999 2005 2008Time 10,0

610,0

09,86 9,79 9,74 9,72

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5 c)

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6)Year Actual

time(s)

Value of the regression function

Linear QuadraticExponentia

l

1960 10,0 10,06 10,03 10,061968 9,95 10,00 10,01 10,001988 9,92 9,86 9,90 9,861999 9,79 9,79 9,80 9,792005 9,77 9,74 9,73 9,742008 9,69 9,72 9,70 9,72

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7)

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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.

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TASK 2

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1)

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2)a)y = 69,14 + 0,06 x

b)y = 75,39 . (1,0004)x

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3)

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4)Distanc

e

Measured

speed

Linear model

Exponential model

1 000 m ~ 145 m/s 127,55 m/s

112,4 m/s

2 700 m ~ 233 m/s 226,85 m/s

220,25 m/s

3 700 m ~ 274 m/s 285,26 m/s

327,62 m/s

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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.