Question

emma recently purchased a new car. she decided to keep track of how many gallons of gas she used on five of her business trips. the results are shown in the table below.

miles driven	number of gallons used
200	10
400	19
600	29
1000	51

round your answer to the nearest hundredth. what is the y-intercept of this equation?

Explanation:

Step1: Define Variables

Let \( x \) be the miles driven (independent variable) and \( y \) be the number of gallons used (dependent variable). We assume a linear relationship \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step2: Calculate the Slope \( m \)

We can use two points to calculate the slope. Let's take the first two points \((x_1,y_1)=(150,7)\) and \((x_2,y_2)=(200,10)\). The formula for the slope is \( m=\frac{y_2 - y_1}{x_2 - x_1}\)
\( m=\frac{10 - 7}{200 - 150}=\frac{3}{50}=0.06\)

We can also use the formula for the slope of a linear regression (since we have multiple points) \( m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\)

First, calculate the necessary sums:
- \( n = 5 \)
- \( \sum_{i=1}^{5}x_i=150 + 200+400 + 600+1000=2350 \)
- \( \sum_{i=1}^{5}y_i=7 + 10+19 + 29+51=116 \)
- \( \sum_{i=1}^{5}x_iy_i=(150\times7)+(200\times10)+(400\times19)+(600\times29)+(1000\times51)=1050+2000 + 7600+17400+51000=79050 \)
- \( \sum_{i=1}^{5}x_i^2=(150^2)+(200^2)+(400^2)+(600^2)+(1000^2)=22500+40000 + 160000+360000+1000000=1582500 \)

Now, substitute into the slope formula:
\( m=\frac{5\times79050-2350\times116}{5\times1582500-(2350)^2}\)
\(=\frac{395250 - 272600}{7912500-5522500}\)
\(=\frac{122650}{2390000}\approx0.0513\) (using linear regression gives a more accurate slope)

Step3: Calculate the y - intercept \( b \)

We use the formula \( b=\bar{y}-m\bar{x}\), where \( \bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\) and \( \bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}\)

\( \bar{x}=\frac{2350}{5} = 470\)
\( \bar{y}=\frac{116}{5}=23.2\)

Using the linear regression slope \( m\approx0.0513\)
\( b=23.2-0.0513\times470\)
\(=23.2 - 24.111\)
\(=- 0.911\approx - 0.91\)

If we use the slope from the first two points \( m = 0.06\)
\( b=7-0.06\times150=7 - 9=- 2\) (but linear regression is more accurate for multiple points)

Using the linear regression method (more appropriate for 5 points):

We can also use the equation of the line \( y=mx + b\), and solve for \( b\) using one of the points. Let's use the point \((150,7)\) and \( m\approx0.0513\)

\( 7=0.0513\times150 + b\)
\( 7 = 7.695+b\)
\( b=7 - 7.695=- 0.695\approx - 0.70\) (Wait, there was a miscalculation in the linear regression slope earlier)

Let's recalculate the slope correctly:

\( n = 5 \)

\( \sum x=150 + 200+400+600+1000 = 2350\)

\( \sum y=7 + 10+19+29+51=116\)

\( \sum xy=150\times7 + 200\times10+400\times19+600\times29+1000\times51=1050 + 2000+7600+17400+51000 = 79050\)

\( \sum x^{2}=150^{2}+200^{2}+400^{2}+600^{2}+1000^{2}=22500 + 40000+160000+360000+1000000 = 1582500\)

\( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}=\frac{5\times79050-2350\times116}{5\times1582500 - 2350^{2}}\)

\( 5\times79050 = 395250\)

\( 2350\times116=2350\times(100 + 16)=235000+37600 = 272600\)

\( 5\times1582500=7912500\)

\( 2350^{2}=(2000 + 350)^{2}=2000^{2}+2\times2000\times350+350^{2}=4000000+1400000 + 122500=5522500\)

\( m=\frac{395250-272600}{7912500 - 5522500}=\frac{122650}{2390000}\approx0.0513\) (this is correct)

\( \bar{x}=\frac{2350}{5}=470\), \( \bar{y}=\frac{116}{5} = 23.2\)

\( b=\bar{y}-m\bar{x}=23.2-0.0513\times470\)

\( 0.0513\times470 = 0.0513\times(400+70)=0.0513\times400+0.0513\times70 = 20.52+3.591 = 24.111\)

\( b=23.2 - 24.111=- 0.911\approx - 0.91\)

But if we use a calculator for linear regression (using all 5 points):

The linear regression equation is of the form \( y = ax + b\), where \( a\) is the slope and \( b\) is the y - intercept.

Using a calculator or software to perform linear regression on the points \((150,7),(200,10),(400,19),(600,29),(1000,51)\):

The slope \( a\approx0.0513\) and the y - intercept \( b\approx - 0.91\)

Answer:

\(-0.91\)