Question

number of raisins in a single serving box
a raisin packing company needs to know about how many raisins are in a box of raisins according to its weight. ten boxes were weighed and the number of raisins inside the box was counted. here are the results.
weight (in grams) number of raisins
42.3 87
42.7 91
42.8 93
42.4 87
42.6 89
42.4 90
42.3 82
42.5 86
42.7 86
42.5 86
what is the slope, m, of the line of best fit?

Explanation:

Step1: Recall slope - formula for line of best fit

Let \(x\) be the weight and \(y\) be the number of raisins. The formula for the slope \(m\) of the line of best - fit in the least - squares regression is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\), where \(n = 10\) (the number of data points).
First, calculate the necessary sums:
Let \(x=\{42.3,42.7,42.8,42.4,42.6,42.4,42.3,42.5,42.7,42.5\}\) and \(y=\{87,91,93,87,89,90,82,86,86,86\}\).
\(\sum_{i = 1}^{10}x_i=42.3 + 42.7+42.8+42.4+42.6+42.4+42.3+42.5+42.7+42.5 = 425.2\).
\(\sum_{i = 1}^{10}y_i=87 + 91+93+87+89+90+82+86+86+86 = 887\).
\(\sum_{i = 1}^{10}x_i^{2}=42.3^{2}+42.7^{2}+42.8^{2}+42.4^{2}+42.6^{2}+42.4^{2}+42.3^{2}+42.5^{2}+42.7^{2}+42.5^{2}\)
\(=1789.29+1823.29+1831.84+1797.76+1814.76+1797.76+1789.29+1806.25+1823.29+1806.25 = 18079.08\).
\(\sum_{i = 1}^{10}x_iy_i=42.3\times87+42.7\times91+42.8\times93+42.4\times87+42.6\times89+42.4\times90+42.3\times82+42.5\times86+42.7\times86+42.5\times86\)
\(=3680.1+3885.7+3980.4+3688.8+3791.4+3816+3468.6+3655+3672.2+3655 = 37492.2\).

Step2: Substitute into the formula

\(n = 10\).
\(n\sum_{i = 1}^{n}x_iy_i=10\times37492.2 = 374922\).
\(\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=425.2\times887=377152.4\).
\(n\sum_{i = 1}^{n}x_i^{2}=10\times18079.08 = 180790.8\).
\((\sum_{i = 1}^{n}x_i)^{2}=425.2^{2}=180795.04\).
\(m=\frac{374922-377152.4}{180790.8 - 180795.04}=\frac{-2230.4}{-4.24}=526.0377\approx526\) (rounded to the nearest whole number).

Answer:

526