Answer :
To determine which equation best models the given data set, let's examine each option step-by-step and compare their residuals.
1. Equation A: [tex]\( y = 2x + 17 \)[/tex]
- Here, for each [tex]\( x \)[/tex] value given in the table, compute [tex]\( y \)[/tex] using the equation [tex]\( y = 2x + 17 \)[/tex].
- Then, compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values from the table and calculate the sum of the squared differences (residuals) between the actual and computed [tex]\( y \)[/tex] values.
2. Equation B: [tex]\( y = 11\sqrt{x - 0.3} + 4.3 \)[/tex]
- For each [tex]\( x \)[/tex] value given in the table, compute [tex]\( y \)[/tex] using the equation [tex]\( y = 11\sqrt{x - 0.3} + 4.3 \)[/tex].
- Compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values from the table and calculate the sum of the squared differences (residuals).
3. Equation C: [tex]\( y = 2x - 17 \)[/tex]
- For each [tex]\( x \)[/tex] value, compute [tex]\( y \)[/tex] using [tex]\( y = 2x - 17 \)[/tex].
- Compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values and calculate the sum of the squared residuals.
4. Equation D: [tex]\( y = 11\sqrt{x + 0.3} - 4.3 \)[/tex]
- For each [tex]\( x \)[/tex] value, compute [tex]\( y \)[/tex] using [tex]\( y = 11\sqrt{x + 0.3} - 4.3 \)[/tex].
- Compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values and calculate the sum of the squared residuals.
By comparing the sum of the squared residuals for each model, the one with the smallest residuals will be the best fit model. This process involves calculations for each equation using the provided values and comparing the accuracy.
After performing this comparison, we find that the equation [tex]\( \boxed{y = 11\sqrt{x - 0.3} + 4.3} \)[/tex] (option B) results in the smallest sum of squared residuals and, hence, best models the provided data set.
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
1. Equation A: [tex]\( y = 2x + 17 \)[/tex]
- Here, for each [tex]\( x \)[/tex] value given in the table, compute [tex]\( y \)[/tex] using the equation [tex]\( y = 2x + 17 \)[/tex].
- Then, compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values from the table and calculate the sum of the squared differences (residuals) between the actual and computed [tex]\( y \)[/tex] values.
2. Equation B: [tex]\( y = 11\sqrt{x - 0.3} + 4.3 \)[/tex]
- For each [tex]\( x \)[/tex] value given in the table, compute [tex]\( y \)[/tex] using the equation [tex]\( y = 11\sqrt{x - 0.3} + 4.3 \)[/tex].
- Compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values from the table and calculate the sum of the squared differences (residuals).
3. Equation C: [tex]\( y = 2x - 17 \)[/tex]
- For each [tex]\( x \)[/tex] value, compute [tex]\( y \)[/tex] using [tex]\( y = 2x - 17 \)[/tex].
- Compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values and calculate the sum of the squared residuals.
4. Equation D: [tex]\( y = 11\sqrt{x + 0.3} - 4.3 \)[/tex]
- For each [tex]\( x \)[/tex] value, compute [tex]\( y \)[/tex] using [tex]\( y = 11\sqrt{x + 0.3} - 4.3 \)[/tex].
- Compare the computed [tex]\( y \)[/tex] with the actual [tex]\( y \)[/tex] values and calculate the sum of the squared residuals.
By comparing the sum of the squared residuals for each model, the one with the smallest residuals will be the best fit model. This process involves calculations for each equation using the provided values and comparing the accuracy.
After performing this comparison, we find that the equation [tex]\( \boxed{y = 11\sqrt{x - 0.3} + 4.3} \)[/tex] (option B) results in the smallest sum of squared residuals and, hence, best models the provided data set.
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
