2024 JPJC H1 Q9
2024 JPJC H1 Q9
10 marks
The owner of an ice cream shop wants to investigate the relationship between the daily average temperature \(x\), in \({}^\circ \mathrm{C}\) and daily sales \(y\), in dollars. The results for a random sample of \(8\) days are shown in the table.
| Day | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
|---|---|---|---|---|---|---|---|---|
| \(x\) | \(24.2\) | \(22.9\) | \(28.5\) | \(29.4\) | \(33.4\) | \(28.1\) | \(32.6\) | \(27.2\) |
| \(y\) | \(218\) | \(183\) | \(402\) | \(419\) | \(554\) | \(418\) | \(483\) | \(394\) |
- Give a sketch of the scatter diagram for the data, as shown on your calculator.[2]
- Find the product moment correlation coefficient and comment on its value in the context of the data.[2]
- Find the equation of the regression line of \(y\) on \(x\) in the form \(y=mx+c\), giving the values of \(m\) and \(c\) correct to \(2\) decimal places. Explain the meaning of the value of \(m\) in the context of the question. Sketch this line on your scatter diagram.[3]
- Use the equation of your regression line to calculate an estimate of the daily sales on a day with average temperature of \(30{}^\circ \mathrm{C}\). Comment on the reliability of your estimate.[2]
- Comment on whether temperature is the only determining factor for the sales of ice-cream at this shop.[1]
Solution:
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Answer:(i) Scatter shown. (ii) \(r\approx0.966\); strong positive linear correlation. (iii) \(y=33.16x-554.18\); about 33.16 dollars more predicted sales per \(1^\circ\mathrm C\). (iv) 440.66 dollars; reasonably reliable interpolation. (v) No; other factors also affect sales.
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