Question

A set of observations of independent variable \( x \) and the corresponding dependent variable \( y \) is given below. \[ \begin{array}{|c|c|} \hline x & y \\ \hline 5 & 16 \\ 2 & 10 \\ 4 & 13 \\ 3 & 12 \\ \hline \end{array} \] Based on the data, the coefficient \( a \) of the linear regression model \[ y = a + bx \] is estimated as 6.1. The coefficient \( b \) is \(\underline{\hspace{2cm}}\) . (round off to one decimal place)

Hint:

The coefficient \( b \) in linear regression represents the slope of the best-fit line and can be calculated using the formula involving the sums of \( x \), \( y \), \( xy \), and \( x^2 \).

Answer 1

Correct Answer: 1.9

The formula for the slope \( b \) of the linear regression line is:
\[ b = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2} \] where \( n \) is the number of data points. From the data, we calculate the necessary sums:
\[ \sum{x} = 5 + 2 + 4 + 3 = 14, \sum{y} = 16 + 10 + 13 + 12 = 51. \] \[ \sum{xy} = (5 \times 16) + (2 \times 10) + (4 \times 13) + (3 \times 12) = 80 + 20 + 52 + 36 = 188. \] \[ \sum{x^2} = 5^2 + 2^2 + 4^2 + 3^2 = 25 + 4 + 16 + 9 = 54. \] Now, substituting into the formula for \( b \):
\[ b = \frac{4 \times 188 - 14 \times 51}{4 \times 54 - 14^2} = \frac{752 - 714}{216 - 196} = \frac{38}{20} = 1.9. \] Thus, the coefficient \( b \) is \( \boxed{1.9} \).