Question

1. Can you tell a child's shoe size by their age? A chart showing shoe size by age for girls was used to obtain the following data:

x= age	y= shoe size
1	5
2	8
4	10
5	11
7	12

a) Construct a scatter diagram of the data. Make sure to label the axes.

b) Does there appear to be a linear relationship?

c) Calculate the correlation coefficient r. Show all of your work. Use a table to organize your work.

Use the correlation formula: r =

d) What does this value of correlation suggest?
e) Find the least squares line and graph it on the scatter diagram from 1a).
f ) Calculate the coefficient of determination, r 2 and explain what it represents.

Below is a suggested table for parts c) and e):

n∑xy−(∑x)(∑y)
n x2− x2 n y2− y2

∑ (∑ ) ∑ (∑ )

	Computation Table
x		y	x2	y2	xy

∑ x = ∑ y = ∑ x 2 = ∑ y 2 = ∑ xy = ( ∑ x )2= ( ∑ y )2 =

Answer 1

a.

b. As we see that there is increasing trend and also a line can be passed through many points

Hence there appear to be linear relationship

c.

X Values
∑ = 19
Mean = 3.8
∑(X - M_x)² = SS_x = 22.8

Y Values
∑ = 46
Mean = 9.2
∑(Y - M_y)² = SS_y = 30.8

X and Y Combined
N = 5
∑(X - M_x)(Y - M_y) = 25.2

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 25.2 / √((22.8)(30.8)) = 0.951

d. r value suggest that there is strong positive correlation between age and shoe size

e.

Sum of X = 19
Sum of Y = 46
Mean X = 3.8
Mean Y = 9.2
Sum of squares (SS_X) = 22.8
Sum of products (SP) = 25.2

Regression Equation = ŷ = bX + a

b = SP/SS_X = 25.2/22.8 = 1.1053

a = M_Y - bM_X = 9.2 - (1.11*3.8) = 5

ŷ = 1.1053X + 5

f. r=0.951, so r^2=0.9044

Which means 90.44% of variation in shoe size is explained by age.