Refit a logarithmic model after finding a recording error

The human resource department of a company is tasked to determine if there is a relationship between the age and cholesterol level of their staff. The following table shows the information on age, $x$ years, and cholesterol level, $y$ milligrams per deciliter of blood, mg/dL, for a random sample of 10 members of staff.

Age, $x$ (years)	38	64	39	34	25	46	37	26	55	31
Cholesterol level, $y$ (mg/dL)	205	225	206	192	150	213	190	165	220	181

(i) Draw a scatter diagram for the data.1 mark

(a) Explain why, in this context, a quadratic model would probably not be appropriate for long term predictions.1 mark

(b) Explain why the diagram shows that the data may be consistent with the model of the form $y=a+b\ln x$.1 mark

One of the values of $y$ is recorded wrongly. Indicate the corresponding point on your diagram by labelling it P.1 mark

Omitting P, use the data for the remaining workers to answer the following questions.

(ii) Calculate the product moment correlation coefficient between $\ln x$ and $y$.1 mark

(iii) Calculate the least squares estimates of $a$ and $b$, for the model $y=a+b\ln x$.2 marks

(iv) Assume that the value of $x$ at P is correct. Estimate the value of $y$ for this value of $x$.2 marks

(v) If the cholesterol levels in the above table were recorded in micrograms per litre of blood instead, state whether your answer in (ii) will be different.1 mark

(vi) The manager wishes to estimate the cholesterol level for an employee who is 22 years old. Comment on the reliability of the estimate.1 mark