Practice section 5 of 6

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]

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

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

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

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]

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

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

(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]

(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]

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]

(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]

(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]

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]

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

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]

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]

(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]

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]

(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]

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

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]

(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]

(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]

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

(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]

Eight randomly selected students were asked to indicate the number of hours they spent studying Mathematics per week during the June holiday. The corresponding Mathematics Mid-year examination scores marked out of 100 marks of the eight students were recorded. The eight pairs of data are given in the table below.

Time spent ($x$ hours)	2.0	2.4	3.0	3.8	4.8	7.2	8.2	9.4
Mathematics score ($y$)	10	19	$p$	47	64	78	76	84

(i) Given that the regression line of $y$ on $x$ is $y=1.9263+9.6223x$, find the value of $p$. [3]

(ii) Draw a scatter diagram to illustrate the data. Calculate the product moment correlation coefficient between $x$ and $y$. [2]

(iii) Calculate the least squares estimates of $a$ and $b$ for the model $y=a+b\ln x$. Find the product moment correlation coefficient between $\ln x$ and $y$. [2]

(iv) Using your answers to parts (i) and (ii), explain which of $y=1.9263+9.6223x$ or $y=a+b\ln x$ is the better model. [2]

(v) Using the appropriate regression line, estimate the Mathematics Mid-year examination score of a student who spent 5.0 hours studying Mathematics during the June holiday. Comment on the reliability of your estimate. [2]

(a) Sketch a scatter diagram that might be expected for the case when $x$ and $y$ are related approximately by $y=a+b\ln x$, where $a$ is positive and $b$ is negative. In this case, your diagram should include 6 points, approximately equally spaced with respect to $x$, and with all $x$- and $y$-values positive. [1]

(b) The mass, $x$ grams, of a certain substance present in a chemical reaction varies with time, $t$ seconds. Observations at successive times give the data shown in the following table.

Time, $t$	5	15	20	25	30	35	40
Mass, $x$	4.5	7.9	8.7	9.6	10.8	11.3	11.9

(i) Draw a scatter diagram for these values, labelling the axes. [1]

(ii) Calculate the product moment correlation coefficient between $t$ and $x$ and comment on its value in context of the data. [2]

(iii) State, with a reason, the effect on the product moment correlation coefficient found in part (ii) if the mass was recorded in milligrams instead of grams. [1]

(iv) State, with a reason, which least squares regression line should be used to express a possible linear relation between $x$ and $t$. Hence find the equation of a suitable regression line for this model. [2]

(v) Use the equation of your regression line to estimate the mass of the certain substance present at 10 seconds and comment on the reliability of your estimate. [2]