Cambridge International Mathematics

3 A caf ́e manager believes that during April thenumber of people wanting dinneris related to the

temperature at noon. Over a 13 day period, the number of diners and the noon temperature were

recorded.

Temperature (xoC) 18 20 23 25 25 22 20 23 27 26 28 24 22

Number of diners (y) 63 70 74 81 77 65 75 87 91 75 96 82 88

a Find the mean point (x,y).

b Draw a scatter diagram for this data.

c Comment on the correlation between the variables.

d Plot (x,y) on the scatter diagram.

e Draw the line of best fit on the scatter diagram.

f Estimate the number of diners at the cafe when it is April ́

and the temperature is:

i 19 oC ii 29 oC.

The problem with drawing a line of best fit by eye is that the answer will vary from one person to another

and the equation of the line may not be very accurate.

Linear regressionis a formal method of finding a line which best fits a set of data.

We can use technology to perform linear regression and hence find the equation of the line. Most graphics

calculators and computer packages use the method of ‘least squares’ to determine the gradient and the

y-intercept.

THE ‘LEAST SQUARES’ REGRESSION LINE

The mathematics behind this method is generally established in

university mathematics courses.

However, in brief, we find the vertical distances d 1 ,d 2 ,d 3 ,

.... to the line of best fit.

We then add the squares of these distances, giving

d 12 +d 22 +d 32 +::::::

Theleast squares regression lineis the one which makes this

sum as small as possible.

Click on the icon. Use trial and error to try to find the least squares line

of best fit for the data provided in the software.

Consider the following data which was collected by a milkbar owner over ten consecutive days:

Max daily temperature (toC) 29 40 35 30 34 34 27 27 19 37

Number of icecreams sold (N) 119 164 131 152 206 169 122 143 63 208

C LINEAR REGRESSION [11.9]

COMPUTER

DEMO

d 1

d (^2) d
3
d 4
d 5
y
x
Two variable analysis (Chapter 22) 461
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Y:\HAESE\IGCSE01\IG01_22\461IGCSE01_22.CDR Monday, 27 October 2008 2:14:51 PM PETER