Transcript Chapter 18C PowerPointx

Material Taken From:
Mathematics
for the international student
Mathematical Studies SL
Mal Coad, Glen Whiffen, John Owen, Robert Haese,
Sandra Haese and Mark Bruce
Haese and Haese Publications, 2004
Section 18B – Measuring Correlation
Correlation Coefficient with Technology
TI 84
• Turn on the Diagnostics
• Enter the data in L1 and
L2
• LinReg L1, L2
Autograph
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•
•
•
•
•
Open a 2D Graph Page
Data > Enter XY Data Set
Enter your data
Select Show Statistics
Click OK
Click Transfer to Results
Box
• View > Results Box
Example 1
At a father-son camp, the heights of the fathers
and their sons were measured.
a) Draw a scatter plot of the data.
b) Calculate r
c) Describe the correlation between the variables.
Example 2
The Botanical Gardens have been trying out a new
chemical to control the number of beetles
infesting their plants. The results are shown.
Sample
a) Calculate r
b) Describe the correlation
between the variables.
A
B
C
D
E
Quantity of
chemical (g)
Number of
surviving beetles
2
5
6
3
9
11
6
4
6
3
Section 18A – Line of Best Fit (By Eye)
At a tournament, athletes throw a discus. The age and
distance thrown are recorded for each athlete:
Drawing the Line of Best Fit
1. Calculate mean of x values x, and mean of y values
2. Mark the mean point x , y on the scatter plot
3. Draw a line through the mean point that is through
the middle of the data
– equal number of points above and below line


y
Example 3
On a hot day, six cars were left in the sun for various lengths of
time. The length of time each car was left in the sun was recorded,
as well as the temperature inside the car at the end of the period.
Calculate
and y
Draw a scatter plot for the data
Plot the mean point and draw a line of best fit
Predict the temperature of a car which has been
left in the sun for:
i. 35 minutes
ii. 75 minutes
e) Comment on the reliability of your predictions
a)
b)
c)
d)
x
Section 18C – Linear Regression
Scatter Plot
Find a linear equation that best-fits the data.
Best Fit vs. Regression
The problem with
drawing a line of
best fit by eye is
that the line will
vary from one
person to the other.
Least Squares Regression Line
• Consider the set of points below.
• Square the distances and find their sum.
• we want that sum to be small.
Least Squares Regression Formula
y y 
s xy
sx
2
(x  x)
Note: sxy will be given on an exam, while sx is found with the TI 84.
Example 4
Consider the data given:
a) Given sxy ≈ -14.286, find the equation of the
least squares regression line.
b) Estimate the value of y when x = 6.
Comment on the reliability of your estimate.
Some things to note:
• The regression line is used for prediction
purposes.
• The regression line is less reliable when
extended far beyond the region of the data.
Example 5
The table shows the sales for Hancock’s
Electronics established in late 1998.
a) Draw a scatterplot to illustrate this data.
b) Given that sxy ≈ 12.5 find the correlation coefficient, r.
c) Find the equation of the regression line, using the formula.
d) Predict the sales figures for the year 2006, giving your answer
to the nearest $10,000.
e) Comment on the reasonableness of this prediction.
Regression Line with Technology
TI 84
• LinReg(ax +b) L1, L2
• where L1 contains your
independent data.
• and L2 contains your
dependent data
Autograph
• Open a 2D Graph Page
• Data > Enter XY Data Set
• Enter your data
• Select Show Statistics
• Click OK
• Click Transfer to Results
Box
• View > Results Box
Example 6
The table shows the annual income and average
weekly grocery bill for a selection of families
a) Construct a scatter plot to illustrate the data.
b) Use technology to find the line of best fit.
c) Estimate the weekly grocery bill for a family
with an annual income of £95000.
Comment on whether this estimate is likely to be reliable.
Homework
• Complete Worksheet 2
• Pg 590 #6,7,8