Question

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.
round your answers to the nearest thousandth.
y = x +

Explanation:

Step 1: Calculate the necessary sums

First, we need to find the following sums: \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \).

Given the data:

• \( x_1 = 11.38, y_1 = 155 \)
• \( x_2 = 19.09, y_2 = 289 \)
• \( x_3 = 22.09, y_3 = 372 \)
• \( x_4 = 23.23, y_4 = 286 \)
• \( x_5 = 26.07, y_5 = 306 \)

Calculate \( \sum x \):
\( \sum x = 11.38 + 19.09 + 22.09 + 23.23 + 26.07 = 101.86 \)

Calculate \( \sum y \):
\( \sum y = 155 + 289 + 372 + 286 + 306 = 1408 \)

Calculate \( \sum xy \):
\( (11.38 \times 155) + (19.09 \times 289) + (22.09 \times 372) + (23.23 \times 286) + (26.07 \times 306) \)
\( = 1763.9 + 5517.01 + 8217.48 + 6643.78 + 7977.42 = 30119.59 \)

Calculate \( \sum x^2 \):
\( (11.38^2) + (19.09^2) + (22.09^2) + (23.23^2) + (26.07^2) \)
\( = 129.5044 + 364.4281 + 487.9681 + 539.6329 + 679.6449 = 2191.1784 \)

Step 2: Calculate the slope \( m \)

The formula for the slope \( m \) of the least squares regression line is:
\( m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)
where \( n = 5 \) (number of data points).

Substitute the values:
\( n\sum xy = 5 \times 30119.59 = 150597.95 \)
\( \sum x \sum y = 101.86 \times 1408 = 143418.88 \)
\( n\sum x^2 = 5 \times 2191.1784 = 10955.892 \)
\( (\sum x)^2 = (101.86)^2 = 10375.4596 \)

Now, calculate the numerator and denominator:
Numerator: \( 150597.95 - 143418.88 = 7179.07 \)
Denominator: \( 10955.892 - 10375.4596 = 580.4324 \)

So, \( m = \frac{7179.07}{580.4324} \approx 12.368 \)

Step 3: Calculate the y-intercept \( b \)

The formula for the y-intercept \( b \) is:
\( b = \frac{\sum y - m \sum x}{n} \)

Substitute the values:
\( m \sum x = 12.368 \times 101.86 \approx 1259.81 \)
\( \sum y - m \sum x = 1408 - 1259.81 = 148.19 \)
\( b = \frac{148.19}{5} \approx 29.638 \)

Answer:

The equation of the least squares regression line is \( y \approx 12.368x + 29.638 \)