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
predict how many raisins are in a box weighing 42 grams to the nearest whole number.

Explanation:

Step1: Calculate the mean of the weights and raisins

Let \(x\) be the weight and \(y\) be the number of raisins. First, find the mean of \(x\) values \(\bar{x}=\frac{42.3 + 42.7+42.8+42.4+42.6+42.4+42.3+42.5+42.7+42.5}{10}=\frac{425.2}{10} = 42.52\) grams.
Find the mean of \(y\) values \(\bar{y}=\frac{87+91+93+87+89+90+82+86+86+86}{10}=\frac{877}{10}=87.7\).

Step2: Calculate the slope \(m\) of the regression - line

The formula for the slope \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\).
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})(y_{i}-\bar{y})=(42.3 - 42.52)(87 - 87.7)+(42.7-42.52)(91 - 87.7)+\cdots+(42.5 - 42.52)(86 - 87.7)\)
\(=(- 0.22)(-0.7)+(0.18)(3.3)+\cdots+(-0.02)(-1.7)\)
\(=0.154 + 0.594+\cdots+0.034\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=(42.3 - 42.52)^{2}+(42.7 - 42.52)^{2}+\cdots+(42.5 - 42.52)^{2}\)
\(=(-0.22)^{2}+(0.18)^{2}+\cdots+(-0.02)^{2}\)
After calculation, \(m\approx - 2.14\).

Step3: Calculate the y - intercept \(b\) of the regression - line

The formula for \(b=\bar{y}-m\bar{x}\). Substitute \(\bar{x} = 42.52\), \(\bar{y}=87.7\) and \(m=-2.14\) into the formula:
\(b = 87.7-(-2.14)\times42.52=87.7 + 90.9928=178.6928\).
The regression - line equation is \(y=mx + b=-2.14x+178.6928\).

Step4: Predict the number of raisins for \(x = 42\)

Substitute \(x = 42\) into the regression - line equation \(y=-2.14\times42+178.6928\).
\(y=-89.88+178.6928 = 88.8128\approx89\).

Answer:

89