Question

applying the least-squares method
the table includes points with the x-coordinates representing the area of the solar panels, in square feet, and the y-coordinates representing the average measured kilowatt hours of energy collected per day. using the least-squares method, what is the linear model?
y = dropdownx + dropdown
the table has columns x, y, x², xy with rows:
5, 40, 25, 200
6.5, 62, 42.25, 403
7.75, 51, 60.0625, 395.25
8.25, 61, 68.0625, 503.25
9.6, 68, 92.16, 652.8
σx = 37, σy = 282, σx² = 288, σxy = 2,154
dropdown options for the first box: 0.27, 4.73, 7.63

Explanation:

Step1: Recall the least - squares formula for slope \(m\)

The formula for the slope \(m\) in the least - squares method is \(m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\), where \(n\) is the number of data points. Here, \(n = 5\), \(\sum x=37\), \(\sum y = 282\), \(\sum xy=2154\), and \(\sum x^{2}=288\).

First, calculate the numerator: \(n\sum xy-\sum x\sum y=5\times2154 - 37\times282\)
\(5\times2154=10770\), \(37\times282 = 10434\)
So, the numerator is \(10770-10434 = 336\)

Then, calculate the denominator: \(n\sum x^{2}-(\sum x)^{2}=5\times288-37^{2}\)
\(5\times288 = 1440\), \(37^{2}=1369\)
So, the denominator is \(1440 - 1369=71\)

Now, find the slope \(m\): \(m=\frac{336}{71}\approx4.73\)

Step2: Recall the formula for the y - intercept \(b\)

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

We know that \(m\approx4.73\), \(\sum y = 282\), \(\sum x=37\), and \(n = 5\)

\(m\sum x=4.73\times37\approx175.01\)

\(\sum y - m\sum x=282-175.01 = 106.99\)

\(b=\frac{106.99}{5}\approx21.4\) (We can also use the formula \(b=\bar{y}-m\bar{x}\), where \(\bar{x}=\frac{\sum x}{n}=\frac{37}{5} = 7.4\) and \(\bar{y}=\frac{\sum y}{n}=\frac{282}{5}=56.4\). Then \(b = 56.4-4.73\times7.4=56.4 - 35.002=21.398\approx21.4\))

Answer:

The linear model is \(y = 4.73x+21.4\) (The slope is \(4.73\) and the y - intercept is approximately \(21.4\))

For the slope part of the drop - down (the coefficient of \(x\)): \(4.73\) (corresponding to the option \(4.73\))