It is given that $X$ is the number of times a student is late for school in a year and $Y$

Question

It is given that $X$ is the number of times a student is late for school in a year and $Y$ is the student's performance in the Mathematics Examination. The product moment correlation coefficient of a bivariate sample $(x_i, y_i)$, $i = 1, 2, \dots, n$, for $n$ students is $r$. State, giving a reason, whether each of the following statements is true or false. When the value of $r$ is zero, it can be implied that the variables $X$ and $Y$ are not related.[1] When the value of $r$ is $-1$, it can be implied that late-coming causes poor performance in the subject.[1] The table below shows the daily sales of cups of iced coffee in a week by a shop and the maximum daily temperature. MonTueWedThu FriSatSun Temperature ($^\circ\mathrm{C}$), $t$$30.5$$31.3$$31.9$$34.8$$25.9$$28.5$$29.2$ Daily sales, $y$$95$$101$$115$$79$$81$$86$$88$ Sketch a scatter diagram of $y$ against $t$, labelling the axes clearly. [2] One of the values of $y$ appears to be incorrect. Indicate the corresponding point on your diagram by labelling it $P$. Omitting $P$, find the equation of the least squares regression line of $\frac{1}{y}$ on $t$, and the value of the product moment correlation coefficient between $\frac{1}{y}$ and $t$. Comment on this value. [5] Use an appropriate regression line to give an estimate of the daily sales when the temperature is $20.4^\circ\mathrm{C}$. State, with a reason, whether the estimate is reliable. [2]

Accepted Answer

(a)(i) False: $r=0$ rules out only linear correlation. (ii) False: correlation does not establish causation. (b)(i) See scatter diagram; $P=(34.8,79)$. (ii) $\dfrac1y=-0.000561t+0.0273$, $r=-0.934$; strong negative linear correlation. (iii) $63$ cups; unreliable.

	Mon	Tue	Wed	Thu	Fri	Sat	Sun
Temperature (\(^\circ\mathrm{C}\)), \(t\)	\(30.5\)	\(31.3\)	\(31.9\)	\(34.8\)	\(25.9\)	\(28.5\)	\(29.2\)
Daily sales, \(y\)	\(95\)	\(101\)	\(115\)	\(79\)	\(81\)	\(86\)	\(88\)

2024 RI P2 Q8

2024 RI P2 Q8

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