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Worked solution

Compare a linear trend with a transformed quadratic

In a country, a government agency tracks the number of cases of dengue fever (NN) each month (TT). The number of cases prior to May 2017 are shown in the table below.

TT123456789101112
NN711647584501407350281222197141133129

(i) Sketch a scatter diagram to illustrate the data. marks

(ii) Calculate the value of the product moment correlation coefficient and find the equation of the regression line of NN on TT, in the form N=a+bTN=a+bT, giving the values of aa and bb correct to 3 significant figures. marks

(iii) A statistician suggested that the model N=k(T−30)2+cN=k(T-30)^2+c would be more appropriate. Find the values of kk and cc. State whether you agree, justifying your answer. marks

[2017/RVHS/MYE/J2/10]

CJC · Review problems · Question 9

Answer

r=−0.979r=-0.979; N=728−56.9TN=728-56.9T. For N=k(T−30)2+cN=k(T-30)^2+c, k=1.22k=1.22, c=−330c=-330, transformed r=0.989r=0.989; prefer the quadratic within the observed range.

(i) Plot the monthly counts

Plot all twelve table pairs with month index TT horizontally and number of cases NN vertically. The counts decrease, with the decline becoming less steep near the end; see the comparison sketch.

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(ii) Linear regression

Regress NN on TT. The correlation is r=−0.9794185826≈−0.979r=-0.9794185826\approx-0.979. The line is N=728.4242424−56.89860140TN=728.4242424-56.89860140T, or N=728−56.9TN=728-56.9T (3 s.f.).

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(iii) Transform and compare

Let u=(T−30)2u=(T-30)^2. For T=1,…,12T=1,\ldots,12, the transformed values are 841, 784, 729, 676, 625, 576, 529, 484, 441, 400, 361, 324. Regress NN on uu to get N=1.220470964u−329.9657022N=1.220470964u-329.9657022. Hence k=1.22k=1.22, c=−330c=-330 (3 s.f.). The transformed correlation is 0.9894813265≈0.9890.9894813265\approx0.989. Since 0.989>∣−0.979∣0.989>|-0.979| and the scatter is curved, the quadratic model better represents this observed range, as in the source. This does not establish that it is suitable for long-term extrapolation: near T=30T=30 the fitted model even becomes negative.

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Compare your sketch

Expand diagram

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Scatter diagram for Question 9: Month index T horizontally, Dengue cases N vertically. All 12 pairs from the question table are plotted.
Scatter diagram for Question 9: Month index T horizontally, Dengue cases N vertically. All 12 pairs from the question table are plotted.
Scatter diagram for Question 9: Month index T horizontally, Dengue cases N vertically. All 12 pairs from the question table are plotted.

Common mistakes

Compare absolute correlations when one transformation reverses the direction. A better sample fit does not certify future epidemiological behaviour.

Solution notes

The transformed working table and every twelve input pair are retained. The source agrees with the quadratic fit on the observed data; the student explanation qualifies its long-term behaviour without altering any printed ask or answer.

Read the hint