Question

s.id.b.6: regression 1
1 the table below shows the number of grams of carbohydrates, x, and the number of calories, y, of six different foods.

carbohydrates (x)	calories (y)
9.5	138
10	147
6	88
7	108
4	62

which equation best represents the line of best fit for this set of data?

1. y = 15x
2. y = 0.07x
3. y = 0.1x - 0.4
4. y = 14.1x + 5.8

Explanation:

Step1: Calculate the mean of \(x\) values

\(\bar{x}=\frac{8 + 9.5+10 + 6+7+4}{6}=\frac{44.5}{6}\approx7.42\)

Step2: Calculate the mean of \(y\) values

\(\bar{y}=\frac{120 + 138+147+88+108+62}{6}=\frac{663}{6}=110.5\)

Step3: Calculate the slope \(m\) using the formula \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

Let's calculate the numerator \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\):
\((8 - 7.42)(120 - 110.5)+(9.5 - 7.42)(138 - 110.5)+(10 - 7.42)(147 - 110.5)+(6 - 7.42)(88 - 110.5)+(7 - 7.42)(108 - 110.5)+(4 - 7.42)(62 - 110.5)\)
\(=0.58\times9.5 + 2.08\times27.5+2.58\times36.5+( - 1.42)\times(- 22.5)+(-0.42)\times(-2.5)+(-3.42)\times(-48.5)\)
\(=5.51+57.2+94.17 + 31.95+1.05+165.87\)
\(=355.75\)
Let's calculate the denominator \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\):
\((8 - 7.42)^{2}+(9.5 - 7.42)^{2}+(10 - 7.42)^{2}+(6 - 7.42)^{2}+(7 - 7.42)^{2}+(4 - 7.42)^{2}\)
\(=0.58^{2}+2.08^{2}+2.58^{2}+(-1.42)^{2}+(-0.42)^{2}+(-3.42)^{2}\)
\(=0.3364+4.3264+6.6564 + 2.0164+0.1764+11.6964\)
\(=25.2084\)
\(m=\frac{355.75}{25.2084}\approx14.1\)

Step4: Calculate the y - intercept \(b\) using the formula \(b=\bar{y}-m\bar{x}\)

\(b = 110.5-14.1\times7.42\)
\(b=110.5 - 104.622\)
\(b\approx5.8\)
The equation of the line of best - fit is \(y = 14.1x+5.8\)

Answer:

1. \(y = 14.1x+5.8\)