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, $x$ days	1	2	3	4	5	10	30	50
Daily rate, $y$ dollars	149	119	115	112	109	105	103	101

Calculate the value of the product-moment correlation coefficient.2 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: $y=a+bx$; B: $y=a+bx^2$; C: $y=a+\dfrac bx$; D: $y=a+b\ln x$.4 marks

For the appropriate model, calculate the least squares estimate of $a$ and $b$, and verify that, for the transformed data, the product-moment correlation coefficient is close to 1.3 marks