To analyze the error Loren made, we need to determine the correct way to find the y-intercept [tex]\(b\)[/tex] in the equation of the trend line in the form [tex]\(y = mx + b\)[/tex].
Given:
- Two points: [tex]\((1, 130)\)[/tex] and [tex]\((10, 149)\)[/tex].
- Loren mistakenly used the equation [tex]\(10 = \frac{19}{9}(149) + b\)[/tex].
Let's follow Loren's mistaken approach and compare it with the correct approach.
1. Calculate the correct slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{149 - 130}{10 - 1} = \frac{19}{9}
\][/tex]
2. Using the provided point ([tex]\(10, 149\)[/tex]), let’s find [tex]\( b \)[/tex]:
Using [tex]\(y = mx + b\)[/tex],
[tex]\[
149 = \frac{19}{9}(10) + b \\
b = 149 - \frac{19}{9} \cdot 10
\][/tex]
So,
[tex]\[
b \approx 149 - 21.11 \approx 127.89
\][/tex]
3. Evaluate the provided options:
- First option: She should have solved [tex]\(10 = \frac{9}{19}(149) + b\)[/tex]
[tex]\[
b = 10 - \frac{9}{19} \cdot 149
\][/tex]
[tex]\[
b \approx 10 - 70.58 \approx -60.58
\][/tex]
This result does not fit our scenario and calculation.
- Second option: She should have solved [tex]\(1 = \frac{19}{9}(130) + b\)[/tex]
[tex]\[
b = 1 - \frac{19}{9} \cdot 130
\][/tex]
[tex]\[
b \approx 1 - 273.44 \approx -272.44
\][/tex]
This again is far from any logical solution for our scenario.
- Third option: She should have solved [tex]\(149 = \frac{19}{9}(10) + b\)[/tex]
[tex]\[
b = 149 - \frac{19}{9} \cdot 10
\][/tex]
[tex]\[
b \approx 149 - 21.11 \approx 127.89
\][/tex]
This matches our correct y-intercept calculation precisely.
- Fourth option: She should have solved [tex]\(130 = \frac{9}{19}(1) + b\)[/tex]
[tex]\[
b = 130 - \frac{9}{19} \cdot 1
\][/tex]
[tex]\[
b \approx 130 - 0.47 \approx 129.53
\][/tex]
This option deviates from our actual calculation.
Hence, after evaluating all the provided options, we find that the third option is correct. Loren should have solved:
[tex]\[
149 = \frac{19}{9}(10) + b
\][/tex]
Therefore, the error she made is that she should have solved [tex]\(149 = \frac{19}{9}(10) + b\)[/tex] for [tex]\(b\)[/tex].
