Porisma

Worked solution

Compare two scatters and fit both regression directions

For each dataset, find (i) the correlation coefficient rr and comment on how it relates to the shape of the scatter diagram, (ii) the regression of yy on xx, and (iii) the regression of xx on yy.

(a)

xxyy
35
712
95
1112
1410
1417
1523
2116
2210
2320
2625

(b)

xxyy
0.65
110
215
2.510
2.82.5
3.67.5
42.5
45
415
510

CJC · Lecture · Self-practice

(a)(i) Interpret the first scatter

Answer

r=0.693r=0.693 (3 s.f.).

The eleven paired points have a positive but visibly scattered linear trend. They are not all close to a perfect line, matching a coefficient between 0 and 1. The source prints r=0.693r=0.693; the scatter comment and plot are authored from the exact pairs. The unrounded check is 0.6926559702…0.6926559702\ldots.

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(a)(ii) Regress y on x

Answer

y=4.50+0.639xy=4.50+0.639x.

Use the first coordinates as explanatory and the second as response: Xlist contains xx, Ylist contains yy. The printed coefficients agree with independently calculated intercept 4.504442925…4.504442925\ldots and slope 0.6390977444…0.6390977444\ldots, rounded to 3 s.f.

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(a)(iii) Regress x on y

Answer

x=4.42+0.751yx=4.42+0.751y.

Swap the explanatory and response lists. The printed reverse coefficients agree with intercept 4.421918908…4.421918908\ldots and slope 0.7507025291…0.7507025291\ldots. This is a new least-squares fit, not an algebraic inversion of the preceding equation.

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(b)(i) Interpret the second scatter

Answer

r=−0.0234r=-0.0234 (3 s.f.).

The ten paired observations show almost no overall linear trend. The coefficient is very slightly negative and close to zero; this does not prove statistical independence or rule out every nonlinear relationship. Preserve all three distinct observations with x=4x=4. The source prints −0.0234-0.0234; the scatter explanation and plot are authored. The unrounded check is −0.02338016784…-0.02338016784\ldots.

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(b)(ii) Regress y on x

Answer

y=8.47−0.0748xy=8.47-0.0748x.

The printed line agrees with independently calculated intercept 8.470628230…8.470628230\ldots and slope −0.07478923035…-0.07478923035\ldots. Its very shallow negative gradient matches the near-zero linear correlation.

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(b)(iii) Regress x on y

Answer

x=3.01−0.00731yx=3.01-0.00731y.

The printed reverse line agrees with independently calculated intercept 3.010299003…3.010299003\ldots and slope −0.007308970100…-0.007308970100\ldots. Both negative slopes have the same sign as rr, but neither is a measure of correlation strength on its own.

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Compare your sketch

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All eleven exact dataset(a) pairs and ten exact dataset(b) pairs shown, retaining repeated x values. Dataset(a) has a scattered positive linear trend; dataset(b) has almost zero linear correlation. No invented fitted curve or deleted observations. y axes begin at zero; x axes are cropped.
All eleven exact dataset(a) pairs and ten exact dataset(b) pairs shown, retaining repeated x values. Dataset(a) has a scattered positive linear trend; dataset(b) has almost zero linear correlation. No invented fitted curve or deleted observations. y axes begin at zero; x axes are cropped.
All eleven exact dataset(a) pairs and ten exact dataset(b) pairs shown, retaining repeated x values. Dataset(a) has a scattered positive linear trend; dataset(b) has almost zero linear correlation. No invented fitted curve or deleted observations. y axes begin at zero; x axes are cropped.

Solution notes

Both complete printed datasets, all six numerical answers, and the additional scatter-comment asks retained. Working and scatter explanations authored and checked independently; no Teacher counterpart or printed marks.