Answer to Question #202492 in Statistics and Probability for Leonie Patrocinio

Question

It is believed that there is a relationship between ‘level of information’ and ‘readiness to take a vaccination jab’ among tertiary institution students. To test this relationship, a research team decides to conduct a study using a sample of tertiary students.

Level of information Readiness score

7 7

9 6

9 6

2 3

8 9

5 7

7 6

6 7

8 6

10 9

1. Use the data supplied in a table to manually calculate the correlation coefficient (r). Show all your calculations
2. What does the ‘correlation coefficient’ that you calculated tell you about the direction and strength (magnitude) of the relationship between information and readiness? 
3. Calculate and interpret the coefficient of determination for this relationship

Answer 1

a)

Let x is Level of information, y is Readiness Score.

$r=\frac{n\sum xy-\sum x\sum y}{(n\sum x^2-(\sum x)^2)(n\sum y^2-(\sum y)^2)}$

$\sum xy=49+54+54+6+72+35+42+42+48+90=492$

$\sum x=7+9+9+2+8+5+7+6+8+10=71$

$\sum y=7+6+6+3+9+7+6+7+6+9=66$

$\sum x^2=49+81+81+4+64+25+49+36+64+100=553$

$\sum y^2=49+36+36+9+81+49+36+49+36+81=462$

$r=\frac{10\cdot492-71\cdot66}{(10\cdot553-71^2)(10\cdot462-66^2)}=\frac{234}{489\cdot264}=0.002$

b)

A value of the correlation coefficient is close to zero, that indicates no relationship between the two variables.

The correlation coefficient is greater than zero, so variables "are moving" in the same direction (positive relationship).

c)

The coefficient of determination:

$r^2=0.002^2=4\cdot10^{-6}$

The value is close to zero, that indicates that the model fails to accurately model the data at all.