Compute r and the least-squares line from data

Before reading a regression output, which variable should be entered as x?

Inspect candidate lines for the original six sessions. Move the adjustment and compare the mean point with the recorded observations. The best fit is calculated in the task below

Must a fitted line pass through every recorded point?

Regression from paired data

A least-squares regression uses the explanatory variable as x and the response variable as y. Read the slope b, intercept a and correlation r from the output, report the stated precision and keep the response variable on the left of the equation

1. 1Set the roles Enter the explanatory variable as x and the response variable as y, regardless of the table order \(\displaystyle x=\text{EV},\quad y=\text{RV}\)
2. 2Calculate the slope The least-squares slope is found before the intercept and gives the average response change per one-unit increase in x. \(\displaystyle b=\frac{S_{xy}}{S_{xx}}\)
3. 3Calculate the intercept Use the unrounded slope when finding the intercept from the two means \(\displaystyle a=\bar{y}-b\bar{x}\)
4. 4Report the output Report a, b and r to the stated number of decimal places and retain the sign of r. \(\displaystyle \hat{y}=a+bx,\quad r=-0.6734\ldots\to-0.67\text{ (2 d.p.)}\)

Given values

\(\displaystyle x=(2,4,6,8),\ y=(5,7,8,10),\ S_{xy}=16,\ S_{xx}=20\)

For the paired data x = (2, 4, 6, 8) and y = (5, 7, 8, 10), calculate the least-squares slope first

Given values

\(\displaystyle \bar{x}=5,\ \bar{y}=7.5,\ b=0.8\)

Use the unrounded slope and the two means to calculate the intercept

Using the wrong x-variable. One of these lines is right. Which one?

Fit the six-session line yourself

Use the original paired (allocated minutes,cards completed) records A(1,3),B(2,4),C(3,6),D(4,5),E(5,8),F(6,7). Predict cards from minutes. Select a regression tool, retain all paired rows, and record which range is X and which is Y. Produce the complete fitted equation with coefficients to three significant figures, then with four significant figures, retaining the full stored coefficients

1. 1Verify and draw Independently calculate the means, Sxx,Sxy,Syy, then b and a. Draw the scatterplot and fitted line with labelled axes x=0..7 minutes,y=0..10 cards. Calculate two points on the fitted line to locate it. Do not simply join two recorded observations
2. 2Test direction and rounding Refit minutes from cards, state the new response and compare that fit with algebraically rearranging the original equation. Explain why they differ. Also calculate an intercept after first rounding the original slope to 0.89 and compare with the unrounded result
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: a complete fit and an independent check

The means are x̄=3.5 and ȳ=5.5. Sxx=Syy=17.5 and Sxy=15.5, giving b=31/35 and a=12/5. Thus predicted cards=12/5+(31/35)×minutes. With coefficients to three significant figures, report predicted cards≈2.40+0.886×minutes. At four significant figures, report predicted cards≈2.400+0.8857×minutes. The slope predicts about 0.886 more cards for each additional minute allocated. The intercept predicts 2.4 cards at zero minutes. Keep the exact values for calculations

1. 1Place the computed line At x=1 the line gives 23/7 cards; at x=6 it gives 54/7 cards. Locate these two computed points and draw a straight line through them, also checking the mean point (3.5,5.5). The observed sessions A and F are not these fitted points
2. 2One reproducible technology record Enter minutes in B2:B7 and cards in C2:C7. Check six paired rows. Use =SLOPE(C2:C7,B2:B7), =INTERCEPT(C2:C7,B2:B7), and =CORREL(B2:B7,C2:C7), checking your tool’s argument order. Draw an XY point plot and add the computed straight line. Independently sum centred products/squares and verify r=31/35. \(\displaystyle \begin{aligned}S_{xx}&=6.25+2.25+0.25+0.25+2.25+6.25=17.5\\S_{yy}&=6.25+2.25+0.25+0.25+6.25+2.25=17.5\\S_{xy}&=6.25+2.25-0.25-0.25+3.75+3.75=15.5\end{aligned}\)
3. 3Direction is part of the fit Refitting minutes from cards gives predicted minutes=−48/35+(31/35)×cards. Algebraically inverting the original fitted equation gives minutes=−84/31+(35/31)×predicted cards, which is different. Swapping lists preserves r, but changes which vertical errors least squares minimises. Here the two fitted slopes happen to match because the spreads match; their product is r²=961/1225, not 1.
4. 4Expose premature rounding Using b=0.89 first gives a=5.5−0.89×3.5=2.385, which rounds to 2.39 at two decimal places. Using b=31/35 gives a=2.4, reported as 2.40 at two decimal places. Retain stored coefficients, then round the final report

What is represented by the straight line on this plot?

Given values

\(\displaystyle S_{xy}=15.5,\ S_{xx}=17.5\)

Calculate the exact fitted slope for cards from minutes

Given values

\(\displaystyle \bar{x}=3.5,\ \bar{y}=5.5,\ b=31/35\)

Calculate the intercept from the exact slope and report three significant figures as a decimal

Given values

\(\displaystyle b=31/35\)

Report the fitted slope to three significant figures as a decimal

Given values

\(\displaystyle a=12/5\)

Report the exact intercept to four significant figures as a decimal

Given values

\(\displaystyle b=31/35\)

Report the exact slope to four significant figures as a decimal

Given values

\(\displaystyle \hat{y}=12/5+(31/35)x,\quad x=1\)

Find the exact fitted response at x=1 to help draw the line

A table lists cards before minutes. What belongs in the explanatory range for predicting cards?

Does solving the fitted cards equation for minutes give the least-squares regression of minutes on cards?

Given values

\(\displaystyle \bar{x}_{new}=5.5,\ \bar{y}_{new}=3.5,\ b_{new}=31/35\)

Calculate the intercept for refitting minutes from cards