Questions: √(ŷ) = 1.4 + 0.6x

Solution Steps

The means of the independent variable \( x \) and the dependent variable \( y \) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 3.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 3.2 \]

The correlation coefficient \( r \) is found to be:

\[ r = 1.0 \]

This indicates a perfect positive linear relationship between \( x \) and \( y \).

The slope \( \beta \) is calculated using the following formulas:

Numerator for \( \beta \):

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 54.0 - 5 \cdot 3.0 \cdot 3.2 = 6.0 \]

Denominator for \( \beta \):

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 55 - 5 \cdot 3.0^2 = 10.0 \]

Thus, the slope \( \beta \) is:

\[ \beta = \frac{6.0}{10.0} = 0.6 \]

The intercept \( \alpha \) is calculated as follows:

\[ \alpha = \bar{y} - \beta \bar{x} = 3.2 - 0.6 \cdot 3.0 = 1.4 \]

The line of best fit can be expressed as:

\[ y = 1.4 + 0.6x \]

Final Answer