Question

1 han creates a scatter plot that displays the relationship between the number of items sold, ( x ), and the total revenue, ( y ), in dollars. han creates a line of best fit and finds that the residual for the point ( (13,930) ) is -40. interpret the meaning of -40 in the context of the problem.
2 the line of best fit for a data set is ( y = 1.1x + 3.4 ). find the residual for each of the coordinate pairs, ( (x,y) ).
a. ( (5,8.8) )
b. ( (2.5,5.95) )
c. ( (0,3.72) )
d. ( (1.5,5.05) )
e. ( (-3,0) )
f. ( (-5,-4.86) )

Explanation:

Step1: Recall the formula for residual

The formula for residual is \( \text{Residual}=y - \hat{y}\), where \(y\) is the observed value and \(\hat{y}\) is the predicted value. The predicted value \(\hat{y}\) is found by substituting \(x\) into the line - of - best - fit equation \(y = 1.1x+3.4\).

Step2: Calculate the predicted value for part (a)

For the point \((x = 5,y = 8.8)\), substitute \(x = 5\) into \(y=1.1x + 3.4\).
\(\hat{y}=1.1\times5+3.4=5.5 + 3.4=8.9\)
Then, use the residual formula: \(\text{Residual}=y-\hat{y}\)
\(\text{Residual}=8.8 - 8.9=-0.1\)

Step3: Calculate the predicted value for part (b)

For the point \((x = 2.5,y = 5.95)\), substitute \(x = 2.5\) into \(y = 1.1x+3.4\).
\(\hat{y}=1.1\times2.5+3.4=2.75+3.4 = 6.15\)
Then, use the residual formula: \(\text{Residual}=y-\hat{y}\)
\(\text{Residual}=5.95 - 6.15=-0.2\)

Step4: Calculate the predicted value for part (c)

For the point \((x = 0,y = 3.72)\), substitute \(x = 0\) into \(y = 1.1x+3.4\).
\(\hat{y}=1.1\times0+3.4=3.4\)
Then, use the residual formula: \(\text{Residual}=y-\hat{y}\)
\(\text{Residual}=3.72 - 3.4 = 0.32\)

Step5: Calculate the predicted value for part (d)

For the point \((x = 1.5,y = 5.05)\), substitute \(x = 1.5\) into \(y = 1.1x+3.4\).
\(\hat{y}=1.1\times1.5+3.4=1.65+3.4=5.05\)
Then, use the residual formula: \(\text{Residual}=y-\hat{y}\)
\(\text{Residual}=5.05 - 5.05=0\)

Step6: Calculate the predicted value for part (e)

For the point \((x=-3,y = 0)\), substitute \(x=-3\) into \(y = 1.1x+3.4\).
\(\hat{y}=1.1\times(-3)+3.4=-3.3 + 3.4=0.1\)
Then, use the residual formula: \(\text{Residual}=y-\hat{y}\)
\(\text{Residual}=0 - 0.1=-0.1\)

Step7: Calculate the predicted value for part (f)

For the point \((x=-5,y=-4.86)\), substitute \(x=-5\) into \(y = 1.1x+3.4\).
\(\hat{y}=1.1\times(-5)+3.4=-5.5+3.4=-2.1\)
Then, use the residual formula: \(\text{Residual}=y-\hat{y}\)
\(\text{Residual}=-4.86-(-2.1)=-4.86 + 2.1=-2.76\)

Answer:

a. \(-0.1\)
b. \(-0.2\)
c. \(0.32\)
d. \(0\)
e. \(-0.1\)
f. \(-2.76\)