Questions: Find the equation for the least squares regression line of the data described below. The design team at an electronics company is evaluating its new prototype for a miniature recording device. As part of this evaluation, designers at the company gathered data about competing devices already on the market. Among other things, the designers recorded the thickness of each recording device (in millimeters), x, and its maximum recording length (in minutes), y. Thickness (in millimeters) Recording time (in minutes) ------ 11.92 136 20.30 117 20.41 119 24.61 118 25.93 478 28.85 476 29.69 396 Round your answers to the nearest thousandth. y = □ x + □

Solution Steps

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

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 23.101 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 262.857 \]

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

\[ r = 0.72 \]

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

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 47213.37 - 7 \cdot 23.101 \cdot 262.857 = 4706.741 \]

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

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 3962.58 - 7 \cdot 23.101^2 = 226.848 \]

The slope \( \beta \) is then computed as:

\[ \beta = \frac{4706.741}{226.848} = 20.748 \]

The intercept \( \alpha \) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 262.857 - 20.748 \cdot 23.101 = -216.461 \]

The equation of the least squares regression line is:

\[ y = -216.461 + 20.748x \]

Final Answer