Residuals and what least squares minimises

Which calculation gives the residual of a point from a fitted line?

Predict whether a zero signed residual sum identifies the best line. Move the slope adjustment through its range, including 0. Every candidate shown passes through the mean point; compare the squared totals and vertical segments

What distinguishes the best line among these candidates?

Residuals and least squares

For a data point, the residual is the actual response value minus the value predicted by the fitted line at the same explanatory value. Least squares chooses the line that minimises the sum of squared residuals

1. 1Predict at the same explanatory value Substitute the point's explanatory value into the fitted line to find its predicted response \(\displaystyle \hat{y}=a+b\left(x_i\right)\)
2. 2Subtract predicted from actual Calculate the residual as the actual response value minus the predicted response value \(\displaystyle e_i=y_i-\hat{y}_i\)
3. 3Keep the sign A positive residual is above the fitted line and a negative residual is below it \(\displaystyle e_i>0\Rightarrow\text{above the line},\quad e_i<0\Rightarrow\text{below the line}\)
4. 4Square before adding Least squares minimises the sum of squared residuals so positive and negative residuals cannot cancel \(\displaystyle \text{least squares: minimise }\sum e_i^{2}\)

Given values

\(\displaystyle \hat{y}=6+2x,\quad x=5\)

For the fitted line ŷ = 6 + 2x and the point (5, 14), first find the predicted response

Given values

\(\displaystyle y=14,\quad\hat{y}=16\)

Use the actual response 14 and the predicted response 16 to find the signed residual

The actual response is 14 and the predicted response is 16. Which calculation gives the signed residual?

Compare three candidate lines

Use the six fictional (minutes,cards) sessions A(1,3),B(2,4),C(3,6),D(4,5),E(5,8),F(6,7). With time as explanatory, compare line A: ŷ=2+x, line B: ŷ=12/5+(31/35)x, and line C: ŷ=5.5. For every case and line, calculate prediction, signed residual and squared residual. Draw the six points and each line, retaining the same axes

1. 1Decide from evidence Calculate Σe and Σe² for each candidate. Identify the smallest squared-error total and explain why the signed total alone cannot choose a line. Explain the signs for sessions C and D in cards, including which prediction is too high
2. 2Choose and verify technology Keep the six paired rows. Choose an XY plot and a spreadsheet or calculator with stored coefficients. Use x=0..7 minutes,y=0..10 cards. Record formula/range selections; independently calculate at least two residuals and the total. Keep exact fractions or unrounded stored values
3. 3Save your work before opening the model Keep the labelled plot, calculations, complete equation, interpretation and tool record. Independent writing, drawing and external-tool execution are not automatically marked; use self or teacher review

Model response: cancellation does not choose the fit

For A, residuals are 0,0,1,−1,1,−1, so Σe=0 and Σe²=4 cards squared. For B, residuals in session order are (−10,−6,33,−33,41,−25)/35; their sum is 0 and Σe²=132/35≈3.7714 cards squared. For C, residuals are −2.5,−1.5,0.5,−0.5,2.5,1.5; their sum is 0 and Σe²=17.5. B has the smallest total of these three candidates

1. 1Explain signs and units For B at C, prediction=177/35 cards and actual=6, so e=33/35 cards: the line underpredicts. At D, prediction=208/35 cards and actual=5, so e=−33/35 cards: the line overpredicts. Residuals have response units; squared residuals have squared response units
2. 2Verify the workflow Put minutes in B2:B7 and cards in C2:C7. In each candidate’s prediction column use its coefficients and the row’s B value; calculate C minus prediction, then residual squared. Sum the last two columns independently. For B the squared numerators sum to 100+36+1089+1089+1681+625=4620; divide by 1225 to obtain 132/35.

Given values

\(\displaystyle x=3,\ y=6,\ \hat{y}=12/5+(31/35)x\)

Using line B, calculate the signed residual for session C in cards

Given values

\(\displaystyle x=4,\ y=5,\ \hat{y}=12/5+(31/35)x\)

Using the exact line B, report the signed residual for D to the nearest whole card

Session D has a negative residual. Which statement uses the sign correctly?

Given values

\(\displaystyle \sum e_i^2=\frac{100+36+1089+1089+1681+625}{1225}\)

Calculate line B’s squared residual total exactly, in cards squared