Question

find the equation for the least squares regression line of the data described below.
dylan read in a science magazine that the frequency of cricket chirps varies according to outdoor temperature. he decided to test this by making several one - minute recordings of synchronized crickets near his house.
he wrote down the outdoor temperature (in celsius), x, at the time of recording. afterward, he slowed down the recording to count the number of chirps, y, the crickets had made.
round your answers to the nearest thousandth.
y = □x + □

Explanation:

Step1: Calculate necessary sums

First, we list the data points:

• \( (x_1, y_1) = (30.82, 159) \)
• \( (x_2, y_2) = (30.82, 154) \)
• \( (x_3, y_3) = (33.83, 166) \)
• \( (x_4, y_4) = (37.27, 161) \)
• \( (x_5, y_5) = (38.94, 175) \)

Calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \):

\( \sum x = 30.82 + 30.82 + 33.83 + 37.27 + 38.94 = 171.68 \)

\( \sum y = 159 + 154 + 166 + 161 + 175 = 815 \)

\( \sum xy = (30.82×159) + (30.82×154) + (33.83×166) + (37.27×161) + (38.94×175) \)
\( = 4900.38 + 4746.28 + 5615.78 + 5990.47 + 6814.5 = 28067.41 \)

\( \sum x^2 = (30.82)^2 + (30.82)^2 + (33.83)^2 + (37.27)^2 + (38.94)^2 \)
\( = 950.8724 + 950.8724 + 1144.4689 + 1389.0529 + 1516.3236 = 5951.5892 \)

Step2: 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:
\( n\sum xy = 5×28067.41 = 140337.05 \)
\( \sum x \sum y = 171.68×815 = 140,029.2 \)
\( n\sum x^2 = 5×5951.5892 = 29757.946 \)
\( (\sum x)^2 = (171.68)^2 = 29474.0224 \)

\( m = \frac{140337.05 - 140029.2}{29757.946 - 29474.0224} = \frac{307.85}{283.9236} ≈ 1.084 \)

Step3: Calculate the y-intercept \( b \)

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

Substitute the values:
\( m\sum x = 1.084×171.68 ≈ 186.101 \)
\( \sum y - m\sum x = 815 - 186.101 = 628.899 \)
\( b = \frac{628.899}{5} ≈ 125.780 \)

Answer:

\( y = 1.084x + 125.780 \)