The sales manager of a retailer of KSI sunblock is monitoring the effects of its television advertis

Question

The sales manager of a retailer of KSI sunblock is monitoring the effects of its television advertising campaign. Over the last $7$ weeks, different durations of television air time, $x$ minutes, were used and the corresponding number of sales, $y$, in hundreds of bottles of KSI sunblock, were recorded as follows: $x$ $8$ $11$ $15$ $18$ $20$ $22$ $25$ $y$ $11.1$ $15.2$ $19.9$ $22.8$ $24.1$ $25.0$ $26.1$ Draw a scatter diagram for the data, labelling the axes clearly. [1] Using the scatter diagram, explain if each of the following can model the given data set. (A): $y=a+b{{x}^{2}}$, where $a$ and $b$ are positive constants (B): $y=a+b\ln x$, where $a$ is negative constant and $b$ is positive constant (C): $y=a+\frac{b}{x}$, where $a$ and $b$ are positive constants. [3] Calculate the least squares estimates of $a$ and $b$, and the product moment correlation coefficient for the model that best fits the data set. [2] Give an interpretation of $b$ in the context of this question.[1] The advertising costs $\$2000$/min and each bottle of KSI sunblock yields a profit of $\$20$. Estimate the weekly profit when $10$ mins of TV time is purchased. Comment on the reliability of the estimate. [3]

Accepted Answer

The scatter diagram in (i) shows that as $x$ increases, $y$ increases by decreasing amounts (or at a decreasing rate). This is consistent with model B. Furthermore, $r=0.997$for Model B, which suggests a strong positive linear relationship between $\ln x$ and $y$. Model A is not appropriate as it suggests that as $x$ increases, $y$ increases by increasing amounts (or at an increasing rate). Model C is not appropriate since it suggests that as $x$ increases, $y$decreases, which is not consistent with the scatter diagram in (i). Equation of least square regression line of $y$ on $\ln x$is $y=-17.326+13.698\ln x$; $a=-17.3$, $b=13.7$, $r=0.997$($3$ s.f.) (iii) For every unit increase in $\ln x$, the corresponding increase in sales of bottles of sunblock is 1369. (iv) $\$8420$. Estimate is reliable since $r=0.997\approx 1$ and $x=10$ is within the range of sample data,$8\le x\le 25$.

2015 ACJC P2 Q12

2015 ACJC P2 Q12