<p style="margin-bottom: 0.5rem">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:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$x$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$8$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$11$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$15$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$18$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$20$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$22$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$25$</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$y$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$11.1$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$15.2$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$19.9$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$22.8$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$24.1$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$25.0$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$26.1$</p></td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Draw a scatter diagram for the data, labelling the axes clearly. 	<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Using the scatter diagram, explain if each of the following can model the given data set.</div><div class="list-item-paragraph" style="text-align: center">(A): $y=a+b{{x}^{2}}$,  where&nbsp;$a$ and $b$ are positive constants   </div><div class="list-item-paragraph" style="text-align: center">(B): $y=a+b\ln x$, where&nbsp;$a$ is negative constant and $b$ is positive constant  </div><div class="list-item-paragraph" style="text-align: center">(C): $y=a+\frac{b}{x}$, where&nbsp;$a$ and $b$ are positive constants. <span class="inline-marks">[3]</span></div></li></ol>
<p>Calculate the least squares estimates of $a$ and $b$, and the product moment correlation coefficient for the model that best fits the data set.	<span class="inline-marks">[2]</span></p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Give an interpretation of $b$ in the context of this question.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The advertising costs $\$2000$/min and each bottle of KSI sunblock yields a profit of $\$20$. Estimate the weekly profit when&nbsp;$10$ mins of TV time is purchased. Comment on the reliability of the estimate.	<span class="inline-marks">[3]</span></div></li></ol>

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:

Draw a scatter diagram for the data, labelling the axes clearly. 	

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. 

Calculate the least squares estimates of $a$ and $b$, and the product moment correlation coefficient for the model that best fits the data set.	

Give an interpretation of $b$ in the context of this question.

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.	

Hypothesis Testing

International Baccalaureate Higher Level Mathematics

2015 ACJC P2 Q12

2015 ACJC P2 Q12