Porisma

Worked solution

Choose the regression line before substituting

Eight observed pairs are:

xxyy
37
48
69
1011
1120
1219
1521
1722

It is given that r=0.93r=0.93, the regression of yy on xx is y=1.188x+3.042y=1.188x+3.042, and the regression of xx on yy is x=0.736y−1.009x=0.736y-1.009.

(a) Taking xx as the independent variable, estimate (i) yy when x=14x=14 and (ii) xx when y=11y=11.

(b) Now taking yy as the independent variable, estimate (i) yy when x=14x=14 and (ii) xx when y=11y=11.

(c) With neither variable identified as independent, estimate (i) yy when x=14x=14 and (ii) xx when y=11y=11.

(d) Use an appropriate regression line to estimate xx if the independent variable yy is correctly determined as 30. Explain why this is not a good estimate.

CJC · Lecture · Example 7

(a)(i) Keep x as the explanatory variable

Answer

y≈19.7y\approx19.7.

The source explicitly selects yy on xx for case (a). Substitute x=14x=14 into the given model: y=1.188(14)+3.042=19.674y=1.188(14)+3.042=19.674. Reported here to 3 s.f.; the visible numerical working is blank.

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(a)(ii) Solve the same fixed model backwards

Answer

x≈6.70x\approx6.70.

Because xx is still prescribed explanatory, use the same y=1.188x+3.042y=1.188x+3.042 and solve 11=1.188x+3.04211=1.188x+3.042. Thus x=(11−3.042)/1.188=6.698653…x=(11-3.042)/1.188=6.698653\ldots. This authored blank completion is not the distinct xx-on-yy fitted estimate.

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(b)(i) Keep y as explanatory

Answer

y≈20.4y\approx20.4.

The applicable fit is now xx on yy. The source prints 14=0.736y−1.00914=0.736y-1.009; completing its blank calculation gives y=(14+1.009)/0.736=20.392663…y=(14+1.009)/0.736=20.392663\ldots.

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(b)(ii) Substitute into the reverse fit

Answer

x≈7.09x\approx7.09.

The printed calculation is x=0.736(11)−1.009=7.09x=0.736(11)-1.009=7.09 to 3 s.f. Before rounding it is 7.0877.087. Here yy is the prescribed explanatory input.

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(c)(i) Predict the requested response when roles are not fixed

Answer

Use yy on xx: y≈19.7y\approx19.7.

Neither explanatory role is prescribed, so predicting yy from a supplied xx calls for the yy-on-xx fit. Substitute x=14x=14 to get 19.67419.674. This completes the blank source choice and calculation.

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(c)(ii) Fit the other response for the other task

Answer

Use xx on yy: x≈7.09x\approx7.09.

For a supplied yy and requested xx, use the xx-on-yy fit: x=0.736(11)−1.009=7.087x=0.736(11)-1.009=7.087. This completes the blank source choice and calculation; it differs from case (a)(ii) because the modelling roles differ.

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(d) Identify the out-of-range explanatory value

Answer

x≈21.1x\approx21.1, but this extrapolation is unreliable.

With yy explanatory use x=0.736y−1.009x=0.736y-1.009, giving x=0.736(30)−1.009=21.071x=0.736(30)-1.009=21.071. The numerical completion is authored. The source explicitly notes that 3030 lies outside the observed 7≤y≤227\le y\le22, so this is extrapolation. Check the range of the explanatory input yy, not only the range of the resulting xx. A high sample correlation does not establish the relationship outside that input range.

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Solution notes

All seven asks, eight pairs, both supplied rounded fitted lines and r retained. Source provides selection reasoning, (b)(ii) and the extrapolation reason; blank numerical/choice completions authored. No printed marks; use the equations given rather than silently replacing them with refitted coefficients.