Answer to Question #174989 in Statistics and Probability for Joshia

Question

B. Survey tests on seIf-concept and on leadership skill were administered to student-leaders. Both tests use a 10-point Likert scale with 10 indicating the highest scores for each test. Scores for students on the tests follow:

Student    | A   |  B   |  C  |  D  | E   |  F  |  G | 

Self

-Concept  | 7.1 | 5.6 | 6.8 | 7.8 | 8.3 | 5.4 | 6.3 |

Leader   | 3.4 | 6.0 | 7.8 | 8.8 | 7.0 | 6.5 | 8.3 |

-ship Skill

1. Compute the coeficient of correlation r.

2. Interpret the results in terms of strength and direction of correlation.

3. Find the regression line that will predict the leadership skill if the self-concept score is known.

Answer 1

1.

$$\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{47.3}{7}=6.757143$$

$$\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{47.8}{7}=6.828571$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum_iX_i)^2$$

$$=326.59-\dfrac{(47.3)^2}{7}=6.977143$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum_iY_i)^2$$

$$=345.98-\dfrac{(47.8)^2}{7}=19.574288$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum_iX_i)(\sum_iY_i)$$

$$=324.91-\dfrac{47.3(47.8)}{7}=1.918571$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}$$

$$r=0.164171$$

2.

$0\leq|r|<0.2$ No correlation.

Positive direction of the correlation.

3.

$$Y=n+mX$$

$$m=\dfrac{SS_{XY}}{SS_{XX}}=0.274980$$

$$n=\bar{Y}-m\bar{X}=4.970495$$

$$Y=4.970495+0.274980X$$