2023 CJC P2 Q6
2023 CJC P2 Q6
8 marks
Prelims
The sales manager of a company that sells air conditioning systems presented the data of average daily temperature, \(t\left( ^{\circ }\mathrm{C} \right)\) and sales, \(s\) (in hundred units) at a meeting.
| Average daily temperature, \(t\) (\({}^\circ\mathrm{C}\)) | \(18\) | \(20\) | \(23\) | \(26\) | \(30\) | \(32\) | \(33\) | \(34\) |
|---|---|---|---|---|---|---|---|---|
| Sales, \(s\) (in hundred units) | \(307\) | \(366\) | \(497\) | \(523\) | \(565\) | \(580\) | \(588\) | \(596\) |
- Draw a scatter diagram of these data and calculate the value of the product moment correlation coefficient between \(s\) and \(t\). Comment on whether a linear model would be appropriate, referring both to the scatter diagram and the value of the product moment correlation coefficient found.[3]
The marketing director proposes that the data should be modelled instead by the regression equation \(s=a\ln t+b\), where \(a\) and \(b\) are constants.
- Find the values of \(a\) and \(b\), giving your answers to \(3\) decimal places.[1]
- Calculate the product moment correlation coefficient between \(s\) and \(\ln t\).[1]
- Using parts (a) and (c), explain which is a better model.[1]
- Use the model proposed by the marketing director to estimate the number of units sold when the average daily temperature is \({{38}^{\circ }}\mathrm{C}\) and comment on its reliability.[2]
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Answer:(a) Although the product moment correlation coefficient \({r = 0.947}\) is close to \({+1}\), which suggests a strong, positive, linear relationship, the scatter diagram indicates that as \({t}\) increases, \({s}\) increases at a decreasing rate, i.e. the scatter diagram shows the points appear to lie on a curve rather than a straight line, so a linear model may not model the relationship well. (b) \({a = 436.107}\) (\({3}\) d.p.), \({b = -923.838}\) (\({3}\) d.p.) (c) \({0.968}\) (\({3}\) s.f.) (d) \({s = a\ln t + b}\) is a better model as the \({r}\)-value is closer to \({1}\). (e) When \({t = 38}\), \({s = 662.53655}\). Sales = \({66254}\) units. Not reliable since \({38^\circ\text{C}}\) is out of the data range hence extrapolation was performed.
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