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Worked solution

Choose a model with a limiting daily rate

The daily rate charged by a car-hire firm varies with the length of the hire period. The firm’s brochure gives the following data.

Hire period, xx days12345103050
Daily rate, yy dollars149119115112109105103101

Calculate the value of the product-moment correlation coefficient. marks

Give a sketch of the scatter diagram for the data, as shown on your calculator, and hence

(i) comment on the value of the product-moment correlation coefficient,

(ii) state, with a reason, which of the following models is appropriate.

A: y=a+bxy=a+bx; B: y=a+bx2y=a+bx^2; C: y=a+bxy=a+\dfrac bx; D: y=a+bln⁡xy=a+b\ln x. marks

For the appropriate model, calculate the least squares estimate of aa and bb, and verify that, for the transformed data, the product-moment correlation coefficient is close to 1. marks

[2006/SP/II/7]

CJC · Review problems · Question 1

Answer

rx,y=−0.563r_{x,y}=-0.563; model C; a=99.8a=99.8, b=47.1b=47.1; r1/x,y=0.992r_{1/x,y}=0.992.

Correlation and scatter diagram

Enter all eight pairs without transforming xx. The product-moment coefficient is r=−0.5634407309≈−0.563r=-0.5634407309\approx-0.563. Plot the eight table pairs with hire period on the horizontal axis and daily rate on the vertical axis; see the comparison sketch. The rate drops rapidly initially, then levels off. The relationship is negative and curved; rr is not close to −1-1, so the original coordinates do not give a close negative linear fit.

How did this part go?

(ii) Select the model

Model C, y=a+b/xy=a+b/x, allows the daily rate to approach a finite limiting value aa as x→∞x\to\infty. This matches the levelling-off pattern and the context of a daily hire rate.

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Fit the transformed data

Put u=1/xu=1/x and regress yy on uu. With b=Suy/Suub=S_{uy}/S_{uu} and a=yˉ−buˉa=\bar y-b\bar u, obtain a=99.78746018a=99.78746018 and b=47.07263415b=47.07263415. Thus a=99.8a=99.8, b=47.1b=47.1 (3 s.f.). The correlation of uu and yy is 0.9918587189≈0.9920.9918587189\approx0.992, close to 11.

How did this part go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Scatter diagram for Question 1: Hire period x (days) horizontally, Daily rate y ($) vertically. All 8 pairs from the question table are plotted.
Scatter diagram for Question 1: Hire period x (days) horizontally, Daily rate y ($) vertically. All 8 pairs from the question table are plotted.
Scatter diagram for Question 1: Hire period x (days) horizontally, Daily rate y ($) vertically. All 8 pairs from the question table are plotted.

Common mistakes

A strong curved relationship can have a coefficient far from ±1. The positive transformed correlation is between 1/x and y, not x and y.

Solution notes

The source calls the original relationship “weak negative linear”. The numerical coefficient and curved scatter are retained; strength labels have no universal cutoff.

Read the hint