Question

1. A cognitive psychologist conducted a study of whether familiarity of words (X) predicts the time it takes (in seconds) to press a button indicating whether the word is singular or plural (Y), with all participants being given the same words. Familiarity with these words was rated at a later time on a 7-point scale (with higher numbers indicating more familiarity). The participants' scores were:

X Y

6 0.3

2 1.5

5 0.8

3 1.4

7 0.1

• Figure the correlation coefficient (10 pts.).
• Describe the correlation (5 pts.)
• Figure the linear prediction rule; list the b value, a value and the final formula (5 pts.).
• Predict the time it takes to press a button if the familiarity with the word is 1 (5 pts.).

Answer 1

X	Y	XY	X²	Y²
6	0.3	1.8	36	0.09
2	1.5	3	4	2.25
5	0.8	4	25	0.64
3	1.4	4.2	9	1.96
7	0.1	0.7	49	0.01

Ʃx =	23
Ʃy =	4.1
Ʃxy =	13.7
Ʃx² =	123
Ʃy² =	4.95
Sample size, n =	5
x̅ = Ʃx/n = 23/5 =	4.6
y̅ = Ʃy/n = 4.1/5 =	0.82
SSxx = Ʃx² - (Ʃx)²/n = 123 - (23)²/5 =	17.2
SSyy = Ʃy² - (Ʃy)²/n = 4.95 - (4.1)²/5 =	1.588
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 13.7 - (23)(4.1)/5 =	-5.16

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = -5.16/√(17.2*1.588) =    -0.9873

There is a strong negative correlation between x and y.

Slope, b = SSxy/SSxx = -5.16/17.2 =    -0.3

y-intercept, a = y̅ -b* x̅ = 0.82 - (-0.3)*4.6 =    2.2

Regression equation :   

ŷ = 2.2 + (-0.3) x  

Predicted value of y at x =    1

ŷ = 2.2 + (-0.3) * 1 = 1.9