Fit and write a transformed least-squares equation

The explanatory variable x is squared before fitting. Which lists should be entered for the regression?

Recall the accepted card-data fitting experiment. Move the slope slider towards its best-fit setting. The same least-squares process will be applied to the coordinates after a one-axis transformation

What is minimised by least squares on the fitted coordinates?

Fitting a transformed regression

A transformed regression fits a straight line to the selected transformed pair. Keep the explanatory list first, read the intercept and slope in the stated order, and retain the transformation when the equation is written

1. 1Put the explanatory list first Use the explanatory variable as the x-list and the response variable as the y-list \(\displaystyle x'=\text{transformed EV},\quad y'=\text{transformed RV}\)
2. 2Transform the selected list Change the named axis before fitting and leave the other axis on its original scale \(\displaystyle x'=x^{2},\quad y'=y\)
3. 3Read the coefficients in order Use the calculator output convention stated in the question for the intercept and slope \(\displaystyle a=6,\quad b=2.5\)
4. 4Keep the transform visible Write the fitted line with the actual variable names and attach the transformation to the changed variable \(\displaystyle \widehat{y}=a+b x^{2}\)

Data or rule

\(\displaystyle x=4,\quad u=x^2\)

Calculate the squared explanatory coordinate

Data or rule

\(\displaystyle \hat y=6+2.5u,\quad u=16\)

Use the stated exact displayed equation on the transformed input

The explanatory variable x is squared before fitting, while y is unchanged. Which equation keeps the transformation on the correct variable?

Build and name the actual fitted pair

Define X and Y as the columns submitted to linear regression. For an input square, X=x² and Y=y; for a response logarithm, X=x and Y=log₁₀(y). Technology symbols a/b may mean different things on different devices: inspect whether the command reports a+bX or aX+b. State which number is the intercept and which multiplies X.

1. 1Preserve the observations Copy whole paired rows, form one new column with the selected formula and keep the other original column unchanged. Check domains and verify at least two transformed entries. Use an XY scatterplot with numerical spacing
2. 2Keep full values Store the unrounded coefficients for residuals or later predictions. Round only the requested report. If a question explicitly defines its displayed rounded equation as the model, use that equation exactly; do not switch conventions silently
3. 3Restore variable notation Replace X/Y with the correct named original variable or its transformation. A response transform appears on the left; an input transform on the right. A fitted log-response value is not a prediction of the original response yet

Fit an input-square model from your own table

Original synthetic calibration observations are (setting x,response reading y)=(1,3),(2,6),(3,10),(4,17). The setting is dimensionless and y is in response units. State x as explanatory and y as response. Choose a calculator or spreadsheet, form u=x², fit y on u and write the fitted equation in x and y. Independently draw raw and transformed scatterplots with suitable labelled windows

1. 1Verify before reporting Calculate the two means and centred sums Sᵤᵤ,Sᵤy independently. Check that the fitted line passes through the transformed mean point. Produce all four predictions/residuals and their sum, weighted sum and squared sum. Report a and b to three decimal places, but use the stored full values for residuals
2. 2Explain the scale State what a unit change in u means for the fitted response. Explain why the slope is not a constant response change for each extra raw x-unit, and why the intercept is outside the observed input interval
3. 3Save your complete work first Save the original and transformed tables, labelled plots, chosen tool/functions, unrounded coefficients, checks and contextual explanation before opening the model. Independent drawing, writing and actual external-tool use are not automatically marked; review them yourself or with a teacher

Model response: Fit an input-square model from your own table

Compare each construction and verification with your saved work. Other correctly specified tools or suitable windows are possible

1. 1Data, command and equation The transformed pairs are (1,3),(4,6),(9,10),(16,17). Use the squared-setting column as X and the unchanged response column as Y in a linear-regression command. Means are ū=15/2 and ȳ=9; Sᵤᵤ=129 and Sᵤy=119. Therefore b=119/129 and a=179/86. The exact equation is ŷ=179/86+(119/129)x²; reports to three decimal places are a=2.081 and b=0.922
2. 2Plots and checks Use x=0..5,y=0..20 for the raw scatter, and u=0..17,y=0..20 for the transformed scatter. The transformed fit passes through(7.5,9). Predictions in original row order are 775/258,1489/258,893/86,4345/258. Residuals are (−1,59,−99,41)/258 response units. Their sum and u-weighted sum are zero; SSE=29/129 response-units². These checks use the exact fit, not 0.922 and 2.081
3. 3Interpretation and limits An increase of one in squared setting u corresponds to 119/129 more fitted response units. A raw change x→x+1 changes u by 2x+1, so its fitted increment depends on x. The intercept predicts at raw x=0, outside 1–4. A close fit to four synthetic observations is an illustration, not evidence of a universal calibration law or a guarantee for new settings

Data or rule

\(\displaystyle S_{uu}=129,\quad S_{uy}=119,\quad b=S_{uy}/S_{uu}\)

Compute the exact fitted slope on the transformed coordinates

Data or rule

\(\displaystyle \bar u=15/2,\quad\bar y=9,\quad b=119/129\)

Compute the intercept from the transformed mean point

Data or rule

\(\displaystyle b=119/129\)

Report this slope to three decimal places

Data or rule

\(\displaystyle y=10,\quad u=9,\quad\hat y=179/86+(119/129)u\)

Calculate the residual for the original observation with x=3.

Which point must lie on the least-squares line in this transformed plot?

A different tool reports slope −0.4 and intercept 8 after taking the reciprocal of x as its explanatory coordinate. Which equation preserves that convention?

Fit and verify a response-log model

Original synthetic observations are(x,y)=(1,3),(2,30),(3,30),(4,300), with dimensionless positive response index y and explanatory setting x. Construct v=log₁₀(y), fit v on x and report the coefficients to four decimal places. Produce a transformed scatterplot and all four residuals. State what response the line predicts and what the residual units represent

1. 1Use an exact common offset Write c=log₁₀(3), so the transformed responses are c,c+1,c+1,c+2. Use these exact differences to verify the slope and residuals independently of rounded logarithms
2. 2Generate and test a change Multiply every raw response by 10. Predict and verify which of the intercept, slope and transformed residuals change. Explain why this does not require a second implementation of least squares
3. 3Save your complete work first Save the original and transformed tables, labelled plots, chosen tool/functions, unrounded coefficients, checks and contextual explanation before opening the model. Independent drawing, writing and actual external-tool use are not automatically marked; review them yourself or with a teacher

Model response: Fit and verify a response-log model

Compare each construction and verification with your saved work. Other correctly specified tools or suitable windows are possible

1. 1Fit and equation Keep original x as the explanatory list and use LOG10(y) as the response list. With c=log₁₀(3), x̄=2.5 and v̄=c+1. The centred sums are Sxx=5 and Sxv=3, so b=3/5 and a=c−1/2. Thus v̂=log₁₀(3)−1/2+(3/5)x. Reports to four decimal places are a=−0.0229 and b=0.6000. Use the full common offset for later calculation
2. 2Plot and residual scale Use x=0..5,v=0..3, label the vertical axis log₁₀(y), and plot(1,c),(2,c+1),(3,c+1),(4,c+2). Residuals are−1/10,3/10,−3/10,1/10 on the log-response scale, with sum 0 and SSE=1/5. For row 2, observed v=c+1 and predicted v=c+7/10, giving 3/10. This is neither 0.3 original response units nor a residual to be exponentiated
3. 3Check the changed data Multiplying every original response by 10 adds 1 to each log-response. The fitted intercept rises by 1, while b and every transformed residual remain unchanged. The new intercept is c+1/2≈0.9771. The equation still predicts a logarithmic response; recovering an original-scale prediction requires an inverse transformation, covered next

Data or rule

\(\displaystyle S_{xx}=5,\quad S_{xv}=3,\quad v=\log_{10}(y)\)

Use exact transformed differences to find the response-log slope

Data or rule

\(\displaystyle v=c+1,\quad\hat v=c+7/10,\quad c=\log_{10}(3)\)

Compute this signed residual on the fitted response scale

Here v is the base-10 logarithm of y. What does the model v̂=a+bx directly estimate?

Data or rule

\(\displaystyle a=\log_{10}(3)-1/2\)

After multiplying all original responses by 10, report the new log-response intercept to four decimal places

Which practice preserves the stated full-fit residual convention?