An electrical appliance supplier is investigating the potential impact of monthly advertisement expe

Question

An electrical appliance supplier is investigating the potential impact of monthly advertisement expenditure, denoted as $x$ (in thousands of dollars), on the popular social media platform SnapGram, on their monthly sales profits, denoted as $y$ (in thousands of dollars). The supplier has collected the following data over the course of the past year.

Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Advertisement Expenditure, $x$	$5$	$6.5$	$7$	$6$	$8$	$5.5$	$10$	$7.5$	$8.5$	$9$	$4$	$9.5$
Sales Profits, $y$	$16$	$19$	$19.5$	$17$	$k$	$16.5$	$32.5$	$22$	$26$	$28$	$15.5$	$30$

It is thought that this set of data can be modelled by one of the following equations:

Model A: $y=mx+c$

Model B: $\ln y=ax+b$

Explain the meaning of the value of $m$ in the context of the data for Model A.[1]
It is given that the least squares regression line for Model A is $y=0.3604+3.0194x$, correct to $4$ decimal places. Show that $k=23.5$, correct to $1$ decimal place.[2]
Draw the scatter diagram for the data, labelling the axes clearly.[1]
By calculating the product moment correlation coefficients and using the scatter diagram from part (iii), explain whether Model A or Model B is a more appropriate model for the given set of data.[3]

Use the more appropriate model selected in part (iv) for the rest of this question.

Use a regression line to obtain the estimate for the sales profit if the advertisement expenditure is $\$11,000$ for a particular month.[2]
Comment on the reliability of the estimate obtained in (v).[1]

Accepted Answer

(i) The monthly sales profits is expected to increase by ${m}$ thousands of dollars for every additional thousand dollars spent on the monthly advertisement expenditure. (ii) ${k \approx 23.5}$ (iv) The product moment correlation coefficient for Model ${\text{A} = 0.968}$. The product moment correlation coefficient for Model ${\text{B} = 0.982}$. Since the product moment correlation coefficient between ${x}$ and ${\ln y}$ is closer to 1 as compared with the product moment correlation coefficient between ${x}$ and ${y}$, it suggests that there is a stronger positive linear correlation between ${x}$ and ${\ln y}$. Furthermore, from the scatter diagram, ${y}$ is increasing at an increasing rate as ${x}$ increases, hence Model B will be the more appropriate model. (v) ${y = 35.88754326}$. Hence the estimate for the sales profit is ${\$35,900}$ (or ${\$35,888}$). (v) Since ${x = 11}$ does not fall within the given data range ${4 \leq x \leq 10}$, the estimate obtained by extrapolation will not be reliable.

Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Advertisement Expenditure, \(x\)	\(5\)	\(6.5\)	\(7\)	\(6\)	\(8\)	\(5.5\)	\(10\)	\(7.5\)	\(8.5\)	\(9\)	\(4\)	\(9.5\)
Sales Profits, \(y\)	\(16\)	\(19\)	\(19.5\)	\(17\)	\(k\)	\(16.5\)	\(32.5\)	\(22\)	\(26\)	\(28\)	\(15.5\)	\(30\)

2023 ACJC P2 Q7

2023 ACJC P2 Q7

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