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 y - intercept, b, of the line of best fit?

Explanation:

Step1: Recall the formula for the line of best - fit in linear regression

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. In linear regression, the formulas for the slope $m$ and y - intercept $b$ are based on the means of $x$ and $y$ values, and the covariance and variance of $x$ and $y$. First, let $x$ be the weight (in grams) and $y$ be the number of raisins. Calculate the means of $x$ and $y$.
Let $x_i$ be the individual $x$ - values and $y_i$ be the individual $y$ - values, and $n = 10$ (the number of data points).
$\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$ and $\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$.
$\sum_{i=1}^{10}x_i=42.3 + 42.7+42.8+42.4+42.6+42.4+42.3+42.5+42.7+42.5=425.2$
$\bar{x}=\frac{425.2}{10}=42.52$
$\sum_{i = 1}^{10}y_i=87 + 91+93+87+89+90+82+86+86+86=887$
$\bar{y}=\frac{887}{10}=88.7$

Step2: Calculate the slope $m$

The formula for the slope $m$ in linear regression is $m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$.
We calculate $(x_i-\bar{x})$ and $(y_i - \bar{y})$ for each $i$, then their products and squares.
For example, for the first data - point: $x_1 = 42.3$, $y_1 = 87$
$x_1-\bar{x}=42.3 - 42.52=-0.22$
$y_1-\bar{y}=87 - 88.7=-1.7$
$(x_1-\bar{x})(y_1 - \bar{y})=(-0.22)\times(-1.7) = 0.374$
$(x_1-\bar{x})^2=(-0.22)^2 = 0.0484$
After calculating for all 10 data - points and summing up:
$\sum_{i = 1}^{10}(x_i-\bar{x})(y_i - \bar{y})=- 2.92$
$\sum_{i = 1}^{10}(x_i-\bar{x})^2=0.292$
$m=\frac{-2.92}{0.292}=-10$

Step3: Calculate the y - intercept $b$

We know that the line of best - fit $y = mx + b$ passes through the point $(\bar{x},\bar{y})$. Substitute $\bar{x}$, $\bar{y}$, and $m$ into the equation $y = mx + b$ and solve for $b$.
$\bar{y}=m\bar{x}+b$
$b=\bar{y}-m\bar{x}$
Substitute $\bar{x}=42.52$, $\bar{y}=88.7$, and $m=-10$
$b=88.7-(-10)\times42.52$
$b=88.7 + 425.2$
$b = 513.9$

Answer:

$513.9$