© Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology
T3 Scotland
Statistics
Two Variable Statistics
©Teachers Teaching with Technology (Scotland)
Higher Statistics
2 Variable Statistics
Objectives
After completing this unit you should be able to use the TI-83 to:
• draw scattergraphs
• calculate and assist you in your interpretation of Pearson’s product-moment correlation
coefficient.
• determine the least squares regression line of y on x given by y = ax + b.
• predict values using this regression line and comment on their reliability.
Example
The table below shows the test results for 10 pupils in both Maths and Physics.
Maths
Physics
(i)
(ii)
(iii)
65
60
45
60
40
55
55
70
60
80
50
40
80
85
30
50
70
70
Draw a scattergraph for this data and comment on the relationship observed.
Calculate the Pearson’s product-moment correlation coefficient.
Find the least squares regression line for this data.
Solution to (i)
Fig 1
1.
Enter the data into the Lists on your Ti-83. (Fig 1)
2.
Enter the STAT PLOT Menu and choose PLOT 1.
On this screen switch ON plot 1.
Choose the TYPE of graph to be a scatter graph, the 1st of the 6 icons.
Choose which data set is to be on the x-axis and which on the y-axis by entering the
appropriate list name (L1 and L2).
Finally choose how the data points will be shown on the graph.(Fig 2)
Fig 2
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Higher Still Statistics (2 Variable Statistics)
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65
80
3. In order to draw the graph to an appropriate range on the axes, choose ZOOM 9:ZoomStat.
(Fig 3)
Fig 3
Fig 4
Your scattergraph is drawn. (Fig 4)
The range on the axes has been set automatically by the calculator, you can see what this
range is by pressing the WINDOW button.(Fig 5)
You may also want to adjust the values selected to those of your choice( Fig 6)
Fig 5
Fig 6
4. To see the graph again press GRAPH. Individual points on the graph can be interrogated
using the TRACE button and the cursor arrows (Fig 7)
Fig 7
3
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Higher Still Statistics (2 Variable Statistics)
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Solution (ii)
Pearson’s product-moment correlation coefficient simplifies algebraically to a
more useful form given by:
sxy
r=
=
sxxsyy
1.
∑ xy (∑ x )
∑ x∑ y
n
2
2
∑ x -
n
(∑ y )
2
2
∑ y -
n
The various statistics used in this formula can be obtained on the Ti-83.
C hoose STAT, CALC 2: 2-Var Stats (Fig 8).
Enter the Names of the list holding the Data i.e. L1 and L2, notice the names are
separated by a comma. (Fig 9)
Fig 8
Fig 9
These various statistics are displayed, notice to see them all you must scroll down the screen
(Fig 10)
Fi g 10
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Higher Statistics (2 Variable Statistics)
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2.
The product-moment correlation coefficient, r can now be calculated, either
manually using the appropriate values from the above screen or using the calculator.
In the CATALOG screen scroll down until you reach DiagnosticOn, press enter twice.(Fig 11)
Fig 11
Now choose STAT, CALC 4:LinReg (ax+b). (Fig 12)
Fig 12
Fig 13
Once enter the List names for the data sets. Using a comma to separate the names. (Fig 13)
On pressing enter the display gives: (Fig 14)
Fig 14
3. The value of Pearson’s product-moment correlation coefficient, r, is now seen. In this
example, r = 0.7365.
This would indicate that although there is a positive correlation it is not very strong
3
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The general equation of the least squares regression line of y on x is given by
∑∑
sxy ∑ xy - n
b= =
(∑x )2
sxx
2
x
∑
n
x
y = ax + b
Where
y
and a = y - bx
The calculator has already evaluated a and b
a = 0.7108 and b = 25.196.
They are
i.e.
The Regression line has equation y = 0.7108x + 25.196.
To draw this line on the graph enter these values on the Y= screen manually, or call them up as
follows.
• Choose Y=, (Fig 15)
Fig 15
• Now choose VARS 5:Statistics (Fig 16)
followed by EQ 1:RegEQ.(Fig 17)
Fig 16
Fig17
• This has automatically “called up” the regression equation into Y1=..
• This can be seen by pressing Y=. (Fig18)
.
Fig 18
Fig 19
• To see the regression line on the graph press GRAPH.(Fig 19)
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Higher Statistics (2 Variable Statistics)
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