Question

You find a correlation between enthusiasm for Cookies and happiness of r = .8. For 12 participants, your sum of cookie enthusiasm (i.e., sum of X) was 120, (sum of X-squared = 1,308), and your sum for happiness (sum of Y) was 144 (sum of Y-squared = 1,920).

Based on this information, calculate the regression equation that predicts happiness as a function of cookie enthusiasm.

Make predictions about happiness values for cookie enthusiasm scores of 3 and 10.
Show all relevant work for all components of this problem.

Answer 1

Solution:

Given:

x = cookie enthusiasm

y = happiness

Correlation coefficient = r = 0.8

n = 12

1 = 120

$\sum x^{2} = 1308$

$\sum y = 144$

$\sum y^{2} = 1920$

Calculate the regression equation that predicts happiness as a function of cookie enthusiasm.

y = bo + b1 X2

where

$b_{1} = r \times \frac{s_{y}}{s_{x}}$

$s_{y}=\sqrt{\frac{\sum y^{2}-(\sum y)^{2}/n}{n-1}}$

Sr Στ2 – (Στ)? n η –1

thus

$s_{y}=\sqrt{\frac{\sum y^{2}-(\sum y)^{2}/n}{n-1}}$

$s_{y}=\sqrt{\frac{1920-(144)^{2}/12}{12-1}}$

$s_{y}=\sqrt{\frac{1920-1728 }{11}}$

$s_{y}=\sqrt{\frac{192 }{11}}$

$s_{y}=\sqrt{ 17.454545 }$

and

Sr Στ2 – (Στ)? n η –1

$s_{x}=\sqrt{\frac{1308-(120)^{2}/12}{12-1}}$

$s_{x}=\sqrt{\frac{1308-1200 }{11}}$

$s_{x}=\sqrt{\frac{108 }{11}}$

$s_{x}=\sqrt{ 9.818182 }$

$s_{x}=3.133398$

thus

$b_{1} = r \times \frac{s_{y}}{s_{x}}$

$b_{1} = 0.8 \times \frac{4.177864 }{3.133398 }$

and

$b_{0}=\bar{y}-b_{1}\times \bar{x}$

where

$\bar{y}=\frac{\sum y}{n}=\frac{144}{12} = 12$

$\bar{x}=\frac{\sum x}{n}=\frac{120}{12}=10$

thus

$b_{0}=\bar{y}-b_{1}\times \bar{x}$

$b_{0}=12- 1.066667 \times 10$

$b_{0}=1.33333$

thus the regression equation that predicts happiness as a function of cookie enthusiasm is:

y = bo + b1 X2

$\mathbf{{\color{DarkBlue} y =1.33333 +1.066667 \times x}}$

_{(Round final answer to specified number of decimal places)}

Make predictions about happiness values for cookie enthusiasm scores of 3 and 10.

Let x = 3,

$y =1.33333 +1.066667 \times x$

$y =1.33333 +1.066667 \times 3$

$\hat{y} =4.53333$

$\mathbf{{\color{DarkOrange} \hat{y} =4.53 }}$

_{(Round final answer to specified number of decimal places)}

Let x = 10

$y =1.33333 +1.066667 \times x$

$y =1.33333 +1.066667 \times 10$

$\mathbf{{\color{DarkGreen} \hat{y }=12.00 }}$