Porisma

Worked solution

Transform PSI and recover an additional observation

A study comparing the number of hotspots detected in Indonesia each day and the average 24-hour PSI reading in Singapore was conducted. The results over 7 days were given below:

Number of hotspots (xx)5097128150170189215
Average 24-hour PSI reading (yy)4598160205263341400

(i) Draw a scatter diagram and find the value of the product moment correlation coefficient between xx and yy. Hence comment on the suitability of the linear model. marks

(ii) Can we conclude that an increase in the number of hotspots will lead to an increase in the PSI reading? Explain your answer. mark

(iii) A few alternative models, in the form of w=ax+bw=ax+b, were proposed to describe the relation between xx and yy. State, with a reason, which model is the most appropriate.

Model A: w=ln⁡yw=\ln y; Model B: w=yw=\sqrt y; Model C: w=y2w=y^2. marks

It was decided that the data for an additional day should be included in the study. After inserting the data for the additional day, it was found that the equation of the regression line of ww on xx is w=0.082742x+2.1735w=0.082742x+2.1735 and the equation of the regression line of xx on ww is x=12.004w−25.106x=12.004w-25.106 based on the most appropriate model in part (iii).

(iv) Find the product moment correlation coefficient between ww and xx. mark

(v) Find the average 24-hour PSI reading on this additional day, correct to the nearest integer. marks

In estimating the number of hotspots when the average 24-hour PSI reading was mm, where 100<m<200100<m<200, Tarzan used the regression line of ww on xx for the estimation and Jane used the regression line of xx on ww for the estimation. It was found that both of them had approximately the same answer. Explain why this was observed. mark

[2013/AJC/II/10]

CJC · Review problems · Question 4

Answer

Raw r=0.979r=0.979; no causal conclusion. Model B (w=yw=\sqrt y); new-data r=0.997r=0.997; additional PSI 240240.

(i) Raw scatter and correlation

Plot all seven (x,y)(x,y) pairs, with hotspots horizontally and PSI vertically. The points have a strong increasing association. r=0.9793976898≈0.979r=0.9793976898\approx0.979. As in the source, a linear approximation is plausible within this range; the visible upward curvature motivates comparison with transformed models.

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(ii) Association is not causation

No. Correlation alone does not establish that increasing hotspots causes the increase in Singapore PSI. Other factors, such as wind direction, could affect the reading.

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(iii) Compare transformations

The correlations of xx with ln⁡y\ln y, y\sqrt y and y2y^2 are 0.99233259620.9923325962, 0.99721855790.9972185579 and 0.91394514730.9139451473. Model B has the coefficient closest to 11, so select w=yw=\sqrt y.

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(iv) Use the two regression slopes

The product of the regression slopes is r2r^2. Both slopes are positive, so r=+0.082742(12.004)=0.9966117439≈0.997r=+\sqrt{0.082742(12.004)}=0.9966117439\approx0.997.

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(v) Recover the missing transformed value

Both regression lines pass through (xˉ,wˉ)(\bar x,\bar w). Solving the printed equations gives xˉ=145.5564438\bar x=145.5564438, wˉ=14.21713127\bar w=14.21713127. For the additional PSI YY, Y=8wˉ−(45+98+160+205+263+341+400)=15.47895955\sqrt Y=8\bar w-(\sqrt{45}+\sqrt{98}+\sqrt{160}+\sqrt{205}+\sqrt{263}+\sqrt{341}+\sqrt{400})=15.47895955. Thus Y=239.5981886Y=239.5981886, or 240240 to the nearest integer.

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Explain the two estimates

Because rr is very close to 11, the two fitted lines are nearly the same line in the (x,w)(x,w) plane. Their predictions are therefore close in this range, although regression of xx on ww is the appropriate direction when estimating hotspots from PSI.

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Compare your sketch

Expand diagram

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Scatter diagram for Question 4: Hotspots x horizontally, Average 24-hour PSI y vertically. All 7 pairs from the question table are plotted.
Scatter diagram for Question 4: Hotspots x horizontally, Average 24-hour PSI y vertically. All 7 pairs from the question table are plotted.
Scatter diagram for Question 4: Hotspots x horizontally, Average 24-hour PSI y vertically. All 7 pairs from the question table are plotted.

Common mistakes

Average sqrt(PSI), not PSI, when recovering the additional day. The source radical is essential.

Solution notes

The PDF visibly prints square roots in Model B, the r formula and the mean equation; native text extraction loses them. Restored from page images. Source intermediate rounding gives sqrt(Y)=15.478; unrounded printed-line arithmetic gives 15.47895955, with the same nearest-integer answer 240.

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