The following table shows the mean temperature, x, in degree Celsius, from noon to 5pm over nine days in June and the number of ice cream cones, y, sold at a beach cafe9.

((i))
Draw a scatter diagram for these values, labeling the axes clearly.
((ii))
Calculate the product moment correlation coefficient for the data and comment on its value.
((iii))
The owner of the cafe9 employed a temporary staff to assist him on a day which had a mean temperature of $33.3^\circ\text{C}$ .
1. ((a))
  Use a suitable regression line to estimate the number of ice cream cones sold on that day and comment on the reliability of this estimate.
2. ((b))
  A check of the accounts showed that the temporary staff had reported a sale of 108 ice cream cones on that day. Using your answer to (a) above, explain whether the owner has reasons to suspect that the temporary staff had been dishonest in handling the profits for that day.
((iv))
By calculating the value of the product moment correlation coefficient between $\ln y$ and $x$ , explaining which of $y = a + bx$ or $\ln y = a + bx$ is the better model. Hence, obtain the best estimate of the percentage increase in the number of ice cream cones sold when the temperature increases by $2^\circ\text{C}$ , assuming that the model you have chosen remains valid.

Scatter diagram Correlation coefficient as measure of fit Use of regression line for prediction Transformation to achieve linearity Interpolation and extrapolation 中文

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Worked solution

9((iii))((b))

1. Recall the estimate and its reliability status from (a)

From part (iii)(a) the number of cones was estimated using the regression line of $y$ on $x$ , but the temperature $33.3^\circ\text{C}$ lies outside the range of the data values of $x$ . Hence the estimate is obtained by extrapolation and is therefore not reliable.

2. Compare with the reported figure and draw the conclusion

Comparing the reported sale of $108$ cones with the regression estimate would only be meaningful if the estimate were reliable. Because the estimate is an unreliable extrapolation, any discrepancy with $108$ cannot be used to justify a charge of dishonesty. Therefore the owner does not have a sound reason, based on this estimate alone, to suspect that the temporary staff was dishonest.

Answer

The owner does not have strong reason to suspect dishonesty. The estimate in (a) was obtained by extrapolation (33.3°C is outside the range of the recorded temperatures), so it is unreliable. Since the predicted figure itself cannot be trusted, the difference between it and the reported sale of 108 cones is not a sound basis for concluding that the temporary staff was dishonest.

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The following table shows the mean temperature, x, in degree Celsius, from noon to 5pm over nine days in June and the number of ice cream cones, y, sold at a beach cafe9.

Worked solution

9((iii))((b))

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