Review practice · Section 1 of 2

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

A chemical is sprayed onto various plots of land. After a year the concentration, $x$, of the chemical is measured in each plot, and also the concentration, $y$, of bacteria in the plot. Both $x$ and $y$ are measured in appropriate units. A statistician is given the following data to analyse.

$x$	5.11	6.17	4.71	4.15	3.87	2.72	10.13	5.44
$y$	4.23	3.98	4.81	6.30	7.82	9.45	11.71	4.15

(i) Calculate the correlation coefficient, and the line of regression of $y$ on $x$.3 marks

(ii) Give a sketch of the scatter diagram for the data, as shown on your calculator. Giving a reason, identify a data pair which should be regarded as suspect.3 marks

(iii) Remove the suspect data pair, and calculate the correlation coefficient, and the line of regression of $y$ on $x$, for the revised data.2 marks

(iv) Comment of your answers to part (i) and (iii).2 marks

Two other plots of land, A and B, are to be sprayed with the chemical. The value of $x$ for A is to be 1.2 and the value of $x$ for B is to be 4.95.

For each of these two plots, find the estimated value of $y$, using the line of regression obtained in part (iii), and state, with a reason in each case, whether it would be good practice to regard this value as a valid prediction.4 marks

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

A study comparing the number of hotspots detected in Indonesia each day and the average 24-hour PSI reading in Singapore was conducted. The results over 7 days were given below:

Number of hotspots ($x$)	50	97	128	150	170	189	215
Average 24-hour PSI reading ($y$)	45	98	160	205	263	341	400

(i) Draw a scatter diagram and find the value of the product moment correlation coefficient between $x$ and $y$. Hence comment on the suitability of the linear model.3 marks

(ii) Can we conclude that an increase in the number of hotspots will lead to an increase in the PSI reading? Explain your answer.1 mark

(iii) A few alternative models, in the form of $w=ax+b$, were proposed to describe the relation between $x$ and $y$. State, with a reason, which model is the most appropriate.

Model A: $w=\ln y$; Model B: $w=\sqrt y$; Model C: $w=y^2$.2 marks

It was decided that the data for an additional day should be included in the study. After inserting the data for the additional day, it was found that the equation of the regression line of $w$ on $x$ is $w=0.082742x+2.1735$ and the equation of the regression line of $x$ on $w$ is $x=12.004w-25.106$ based on the most appropriate model in part (iii).

(iv) Find the product moment correlation coefficient between $w$ and $x$.1 mark

(v) Find the average 24-hour PSI reading on this additional day, correct to the nearest integer.3 marks

In estimating the number of hotspots when the average 24-hour PSI reading was $m$, where $100<m<200$, Tarzan used the regression line of $w$ on $x$ for the estimation and Jane used the regression line of $x$ on $w$ for the estimation. It was found that both of them had approximately the same answer. Explain why this was observed.1 mark

A scientist is studying the growth of a bacteria colony in a petri dish. He places bacteria at one spot of the dish and he measures the area $A$ the bacteria cover on day $t$. His results are recorded below:

Time, $t$ (days)	0	1	2	3	4	5	6	7
Area, $A$ ($\mathrm{cm}^2$)	1.00	5.00	27.10	41.39	46.95	49.95	51.07	52.05

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

(ii) The scientist would like to predict the future growth of the bacteria. Explain why, in this context, a linear model is not likely to be appropriate.1 mark

It is proposed to fit a model of the form $\ln(D-A)=a+bt$, where $D$ is a suitable constant. The product moment correlation coefficient between $t$ and $\ln(D-A)$ is denoted by $r$. The following table gives values of $r$ for some possible values of $D$.

$D$	53	54	55
$r$		−0.99132	−0.98767

(iii) Calculate the value of $r$ for $D=53$, giving your answer correct to 5 decimal places. Hence, explain which value of $D$ is the most appropriate.2 marks

(iv) With the value of $D$ in (iii), calculate the values of $a$ and $b$ correct to 5 decimal places, and use them to predict the area covered by the bacteria after 20 days.3 marks

(v) Give an interpretation, in context, of the value of $D$.1 mark

A new drug developed to reduce a smoker’s reliance on nicotine may reduce one’s pulse rate to dangerously low levels. To investigate the drug’s effect on pulse rate, different dosages of the drug were administered to six smokers, and the decrease in each smoker’s pulse rate was recorded 30 minutes later.

Smoker	1	2	3	4	5	6
Drug dosage ($x\ \mathrm{cm}^3$)	2.0	1.5	3.0	2.5	4.0	3.5
Decrease in pulse rate ($y$ beats per minute)	14	9	18	16	21	20

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

(ii) Calculate the product moment correlation coefficient between $x$ and $y$, and explain why its value does not necessarily mean that the best model for the relationship between $x$ and $y$ is $y=a+bx$.2 marks

(iii) It is proposed that the decrease in pulse rate can be modelled by the formula $y=\ln(cx^d)$, where $c$ and $d$ are constants. Explain why this is a better model.1 mark

(iv) Calculate the least-squares estimates of $c$ and $d$ for the model in (iii).2 marks

(v) Use a regression line to give the best estimate that you can of the dosage of drug administered when the decrease in pulse rate is 25 beats per minute. Comment on the reliability of your answer.3 marks