Questions: Question 5 (1 point) What is interpolation in the context of linear regression? Predicting values within the range of the observed data Predicting values outside the range of the observed data Calculating the residual sum of squares Estimating the y-intercept of the line

Solution Steps

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

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

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 6.0 \]

The correlation coefficient \( r \) is determined 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 \( \beta \):

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 110 - 5 \cdot 3.0 \cdot 6.0 = 20.0 \]

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:

\[ \beta = \frac{20.0}{10.0} = 2.0 \]

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

\[ \alpha = \bar{y} - \beta \bar{x} = 6.0 - 2.0 \cdot 3.0 = 0.0 \]

The equation of the line of best fit is given by:

\[ y = 0.0 + 2.0x \]

Interpolation in the context of linear regression refers to predicting values within the range of the observed data. It involves estimating the dependent variable \( y \) for a given independent variable \( x \) that lies within the range of the data points used to fit the model.

Final Answer