Review practice · Section 2 of 2

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 marks

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

(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 marks

(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 marks

(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 marks

(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 mark

(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 mark

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

(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 mark

(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 marks

(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 marks

In a country, a government agency tracks the number of cases of dengue fever ($N$) each month ($T$). The number of cases prior to May 2017 are shown in the table below.

$T$	1	2	3	4	5	6	7	8	9	10	11	12
$N$	711	647	584	501	407	350	281	222	197	141	133	129

(i) Sketch a scatter diagram to illustrate the data.2 marks

(ii) Calculate the value of the product moment correlation coefficient and find the equation of the regression line of $N$ on $T$, in the form $N=a+bT$, giving the values of $a$ and $b$ correct to 3 significant figures.2 marks

(iii) A statistician suggested that the model $N=k(T-30)^2+c$ would be more appropriate. Find the values of $k$ and $c$. State whether you agree, justifying your answer.3 marks