Answer to Question #222098 in Statistics and Probability for Darryl

Question

The following sample observations were randomly selected

X 4 5 3 6 10 

Y 4 6 5 7 7 

a) Determine the correlation coefficient and interpret the relationship between X and Y. 

b) Determine the regression equation 

c) Determine the value of when X is 7. 

d) State two uses of goodness-of-fit test

Answer 1

Solution:

Given:

(a):

$X\ Values: \\∑ = 28 \\Mean = 5.6 \\∑(X - M_x)^2 = SS_x = 29.2 \\Y\ Values: \\∑ = 29 \\Mean = 5.8 \\∑(Y - M_y)^2 = SS_y = 6.8 \\X\ and\ Y\ Combined: \\N = 5 \\∑(X - M_x)(Y - M_y) = 10.6$

$r\ Calculation: \\r = ∑((X - M_y)(Y - M_x)) / \sqrt{(SS_x)(SS_y)} \\r = 10.6 / \sqrt{(29.2)(6.8)} = 0.7522$

So, the correlation coefficient=0.7522

Interpretation: This is a strong positive correlation, which means that high X variable scores go with high Y variable scores (and vice versa).

(b):

$\\Sum\ of\ products (SP) = 10.6 \\Regression\ Equation = ŷ = bX + a \\b = SP/SS_X = 10.6/29.2 = 0.36301 \\a = M_Y - bM_X = 5.8 - (0.36\times5.6) = 3.76712 \\ŷ = 0.36301X + 3.76712$

(c):

Put x=7

$ŷ = 0.36301(7) + 3.76712 \\ŷ =6.30819$

(d):

Uses:

1. The Chi-square goodness of fit test is a statistical hypothesis test used to determine whether a variable is likely to come from a specified distribution or not.
2. It is often used to evaluate whether sample data is representative of the full population.