Shanti wrote the predicted values for a data set using the line of best fit [tex]y=2.55 x-3.15[/tex]. She computed two of the residual values.
| x | Given | Predicted | Residual |
|---|---|---|---|
| 1 | -0.7 | -0.6 | -0.1 |
| 2 | 2.3 | 1.95 | 0.35 |
| 3 | 4.1 | 4.5 | a |
| 4 | 7.2 | 7.05 | b |
What are the values of [tex]a[/tex] and [tex]b[/tex]?
A. [tex]a=0.4[/tex] and [tex]b=-0.15[/tex]
B. [tex]a=-0.4[/tex] and [tex]b=0.15[/tex]
C. [tex]a=8.6[/tex] and [tex]b=14.25[/tex]
D. [tex]a=-8.6[/tex] and [tex]b=-14.25[/tex]
Answer (2)
Calculate the predicted value for x = 3 : y = 2.55 × 3 − 3.15 = 4.5 .
Calculate the residual a for x = 3 : a = 4.1 − 4.5 = − 0.4 .
Calculate the predicted value for x = 4 : y = 2.55 × 4 − 3.15 = 7.05 .
Calculate the residual b for x = 4 : b = 7.2 − 7.05 = 0.15 . Therefore, a = − 0.4 and b = 0.15 , so the answer is a = − 0.4 and b = 0.15 .
Explanation
Understanding the Problem We are given a table with x values, given y values, predicted y values, and residual values. We need to find the values of a and b , which are the residuals for x = 3 and x = 4 respectively. The equation of the line of best fit is y = 2.55 x − 3.15 , and the residual is calculated as Given - Predicted.
Calculating Residual a First, we calculate the predicted value for x = 3 using the equation y = 2.55 x − 3.15 :
y = 2.55 × 3 − 3.15 = 7.65 − 3.15 = 4.5 Then, we calculate the residual a for x = 3 by subtracting the predicted value from the given value: a = 4.1 − 4.5 = − 0.4
Calculating Residual b Next, we calculate the predicted value for x = 4 using the equation y = 2.55 x − 3.15 :
y = 2.55 × 4 − 3.15 = 10.2 − 3.15 = 7.05 Then, we calculate the residual b for x = 4 by subtracting the predicted value from the given value: b = 7.2 − 7.05 = 0.15
Final Answer Therefore, the values of a and b are − 0.4 and 0.15 respectively.
Examples
In data analysis, residuals help us understand how well a line of best fit represents the data. For example, if you're predicting house prices based on size, a residual tells you how much the actual price differs from your prediction. Understanding residuals is crucial in assessing the accuracy of your models and making informed decisions based on data.
The values of the residuals are a = − 0.4 and b = 0.15 . Thus, the correct answer is option B. Residuals help us understand the accuracy of our predictions based on the line of best fit.
;
