Material Taken From:

Mathematicsfor the international student

Mathematical Studies SL

Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce

Haese and Haese Publications, 2004

Correlation Coefficient with Technology

TI 84

• Turn on the Diagnostics• Enter the data in L1 and

L2• LinReg L1, L2

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

Section 18B – Measuring Correlation

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 rc) Describe the correlation between the variables.

Example 1

The Botanical Gardens have been trying out a new chemical to control the number of beetles

infesting their plants. The results are shown.

a) Calculate rb) Describe the correlation

between the variables.

Sample Quantity of chemical (g)

Number of surviving beetles

A 2 11B 5 6C 6 4D 3 6E 9 3

Example 2

At a tournament, athletes throw a discus. The age and distance thrown are recorded for each athlete:

Section 18A – Line of Best Fit (By Eye)

Drawing the Line of Best Fit

1. Calculate mean of x values , and mean of y values2. Mark the mean point on the scatter plot3. Draw a line through the mean point that is through

the middle of the data– equal number of points above and below line

x y ,x y

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.

a) Calculate andb) Draw a scatter plot for the datac) Plot the mean point and draw a line of best fitd) Predict the temperature of a car which has been

left in the sun for:i. 35 minutesii. 75 minutes

e) Comment on the reliability of your predictions

x y

Example 3

Scatter Plot

Find a linear equation that best-fits the data.

Section 18C – Linear Regression

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

Note: sxy will be given on an exam, while sx is found with the TI 84.

)(2 xxss

yyx

xy

Consider the data given:

b) Estimate the value of y when x = 6. Comment on the reliability of your estimate.

a) Given sxy ≈ -14.286, find the equation of the least squares regression line.

Example 4

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.

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.

Example 5

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

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.

Example 6

Homework

• Complete Worksheet 2• Pg 590 #6,7,8