It is believed that there is a correlation between the number of fogging sessions and the number of new dengue cases. To test the effectiveness of fogging on controlling the spread of dengue, a particular neighbourhood is subjected to a number of fogging sessions ( $x$ ) spread out regularly in a month and the corresponding number of new dengue cases ( $y$ ) is recorded. The results are summarised in the following table:

((i))
Calculate the product moment correlation coefficient between $x$ and $y$ , and explain whether your answer suggests that a linear model is appropriate.
((ii))
Draw a scatter diagram for the data.
((iii))
One of the data points appears to be incorrect.

Indicate this point by labelling it as $P$ , and explain why the scatter diagram for the remaining points may be consistent with a model of the form

$y = Ae^{bx}$ .
((iv))
Omitting $P$ , use an appropriate regression line to give the best estimate, to the nearest whole number, of the number of fogging sessions in a month when the number of new dengue cases is 7.

Interpreting product moment correlation coefficient Scatter diagram Use of regression line for prediction Correlation coefficient as measure of fit Transformation to achieve linearity 中文

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Worked solution

8((i))

1. Calculate the product moment correlation coefficient $r$ using a calculator

Entering the values of $x$ and $y$ into a graphing calculator:
$r = -0.95509...$
$r = -0.955$ (to 3 s.f.)

2. Explain whether the value of $r$ suggests a linear model is appropriate

Since $r = -0.955$ is close to $-1$ , it indicates a strong negative linear correlation between $x$ and $y$ . Therefore, based on this value alone, it suggests that a linear model is appropriate.

Answer

$r = -0.955$
Since $r$ is close to $-1$ , it suggests a strong negative linear correlation, meaning a linear model is appropriate.

8((ii))

1. Draw axes, determine scales, and plot the given data points

Set up a pair of axes with appropriate linear scales to accommodate the domains $0 \le x \le 14$ and $0 \le y \le 20$ .
Plot the given points accurately: $(2, 20)$ , $(4, 14)$ , $(6, 9)$ , $(8, 6)$ , $(10, 7)$ , $(12, 2)$ , and $(14, 1)$ .
Label the axes clearly as 'No. of fogging sessions, $x$ ' and 'No. of new dengue cases, $y$ '.

Answer

[A properly scaled scatter diagram with the points $(2, 20)$ , $(4, 14)$ , $(6, 9)$ , $(8, 6)$ , $(10, 7)$ , $(12, 2)$ , and $(14, 1)$ plotted, with the horizontal axis labelled $x$ and the vertical axis labelled $y$ .]

8((iii))

1. Identify the incorrect data point $P$ from the anomaly in the trend.

Observing the general trend, as $x$ increases, $y$ is decreasing. However, from $x=8$ to $x=10$ , $y$ incorrectly increases from $6$ to $7$ . Thus, the anomalous point is $(10, 7)$ .

$P(10, 7)$

2. Explain why the remaining points are consistent with an exponential model.

Excluding $P$ , the remaining points on the scatter diagram form a curve where the gradient is negative but increasing towards $0$ (i.e. becoming less steep). This means $y$ decreases at a decreasing rate as $x$ increases, which visually matches the shape of an exponential curve $y = Ae^{bx}$ with $A > 0$ and $b < 0$ .

Answer

$P$ is the point $(10, 7)$ .

The scatter diagram of the remaining points shows that as $x$ increases, $y$ decreases at a decreasing rate. This non-linear, concave upward decreasing trend is characteristic of an exponential decay curve $y = Ae^{bx}$ where $A > 0$ and $b < 0$ .

8((iv))

1. Identify the appropriate regression line and find its equation using the valid data points.

Since the number of fogging sessions ( $x$ ) is predetermined and spread out regularly, $x$ is the independent (controlled) variable. Therefore, the appropriate regression line to use is the regression line of $\ln y$ on $x$ .

Omitting the anomalous point $P(10, 7)$ , we use the remaining 6 data points to obtain the equation from a graphing calculator:
$\ln y = -0.24888x + 3.6276$

2. Substitute the given value of $y$ to estimate $x$.

Substitute $y = 7$ into the regression equation to find the corresponding value of $x$ :
$\ln 7 = -0.24888x + 3.6276$
$0.24888x = 3.6276 - 1.9459$
$0.24888x = 1.6817$
$x \approx 6.757$

Rounding to the nearest whole number, the best estimate for the number of fogging sessions is $7$ .

Answer

$7$

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Worked solution

8((i))

8((ii))

8((iii))

8((iv))

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