Questions: Answer the quiz question. Lower your LabPad and take a look at the PC screen again to explore the graph. A regression formula is obtained from the standard curve. This formula describes the relationship between concentration and absorbance data. We can use this regression formula to calculate the concentration of unknown samples. In this case, which regression type best describes our absorbance data? a) Linear b) Exponential c) Power d) log -log

Solution Steps

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

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 0.3 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 0.35 \]

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

\[ r = 1.0 \]

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

The numerator for the slope \( \beta \) is calculated as:

\[ \text{Numerator for } \beta = \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 0.625 - 5 \cdot 0.3 \cdot 0.35 = 0.1 \]

The denominator for \( \beta \) is:

\[ \text{Denominator for } \beta = \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 0.55 - 5 \cdot 0.3^2 = 0.1 \]

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

\[ \beta = \frac{0.1}{0.1} = 1.0 \]

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

\[ \alpha = \bar{y} - \beta \bar{x} = 0.35 - 1.0 \cdot 0.3 = 0.05 \]

The line of best fit can be expressed as:

\[ y = 0.05 + 1.0x \]

Given that the correlation coefficient \( r = 1.0 \) indicates a perfect linear relationship, we conclude that the data is best described by a linear regression model.

Final Answer