Standard +0.8 This is a multi-part Further Maths statistics question requiring hypothesis testing for correlation (including critical value lookup for n=5), understanding the relationship between regression coefficients and correlation (using r = ±√(b_yx × b_xy)), and judging reliability of predictions. While the individual techniques are standard, the combination of correlation testing, algebraic manipulation of regression equations through their point of intersection at means, and critical evaluation requires solid understanding beyond routine application.
For a random sample of 5 observations of pairs of values \((x, y)\), the equation of the regression line of \(y\) on \(x\) is \(y = -4.2 + c\) and the equation of the regression line of \(x\) on \(y\) is \(x = 10.8 + dy\), where \(c\) and \(d\) are constants. The product moment correlation coefficient is \(-0.7214\) and the mean value of \(x\) is 7.018.
\begin{enumerate}[label=(\roman*)]
\item Test at the 5% significance level whether there is evidence of non-zero correlation between the variables.
[4]
\item Find the values of \(c\) and \(d\).
[5]
\item Use an appropriate regression line to estimate the value of \(x\) when \(y = 3.5\), and comment on the reliability of your estimate.
[2]
\end{enumerate]
OR: t = r√((n–2) / (1 – r2)) = –2.08, t = 2.776 or 2.78
Answer
Marks
Guidance
r 4,0.975
(*B1
(Rarely seen)
Accept H if
t
< tab. t-value (AEF)
0 r
M1)
No evidence of [non-zero] correlation (AEF)
A1
Correct conclusion (dep *B1)
4
Answer
Marks
Guidance
Question
Answer
Marks
Answer
Marks
Guidance
9(ii)
cd = r2 [= (–0.7214)2 = 0.52042]
M1
x = 10.8 + d (4.2 + cx) [y = 2.137]
M1
Find 2nd eqn. for c, d using means in eqns. of regression lines
Combine to find c (or d)
c = 4.2 r2 / {(1 – r2)x – 10.8} = – 0.294
Answer
Marks
Guidance
or d = {(1 – r2)x – 10.8} / 4.2 = – 1.77
M1A1
d = 0.72142 / c = – 1.77 or c = 0.72142 / d = – 0.294
A1
and hence d (or c)
5
Answer
Marks
Guidance
9(iii)
x = 10.8 – 1.77 × 3.5 = 4.60[5] [y on x gives 2.38]
B1
e.g. not reliable since no evidence of correlation
or reasonably reliable since 0.7214 close to 1
or not reliable since 0.7214 not close to 1
Answer
Marks
Guidance
or reliability unclear as degree of extrapolation unknown
B1
Valid comment on reliability (AEF)
2
Answer
Marks
Guidance
Question
Answer
Marks
Question 9:
--- 9(i) ---
9(i) | H : ρ= 0, H : ρ≠ 0
0 1 | B1 | State both hypotheses (B0 for r …)
EITHER: r = 0.878
5, 5% | *B1 | State or use correct tabular two-tail r-value
Accept H if |r| < tab. value (AEF)
0 | M1 | State or imply valid method for conclusion
OR: t = r√((n–2) / (1 – r2)) = –2.08, t = 2.776 or 2.78
r 4,0.975 | (*B1 | (Rarely seen)
Accept H if | t | < tab. t-value (AEF)
0 r | M1)
No evidence of [non-zero] correlation (AEF) | A1 | Correct conclusion (dep *B1)
4
Question | Answer | Marks | Guidance
--- 9(ii) ---
9(ii) | cd = r2 [= (–0.7214)2 = 0.52042] | M1 | Find cd
x = 10.8 + d (4.2 + cx) [y = 2.137] | M1 | Find 2nd eqn. for c, d using means in eqns. of regression lines
Combine to find c (or d)
c = 4.2 r2 / {(1 – r2)x – 10.8} = – 0.294
or d = {(1 – r2)x – 10.8} / 4.2 = – 1.77 | M1A1
d = 0.72142 / c = – 1.77 or c = 0.72142 / d = – 0.294 | A1 | and hence d (or c)
5
--- 9(iii) ---
9(iii) | x = 10.8 – 1.77 × 3.5 = 4.60[5] [y on x gives 2.38] | B1 | Find y from eqn. of regression line of x on y
e.g. not reliable since no evidence of correlation
or reasonably reliable since 0.7214 close to 1
or not reliable since 0.7214 not close to 1
or reliability unclear as degree of extrapolation unknown | B1 | Valid comment on reliability (AEF)
2
Question | Answer | Marks | Guidance
For a random sample of 5 observations of pairs of values $(x, y)$, the equation of the regression line of $y$ on $x$ is $y = -4.2 + c$ and the equation of the regression line of $x$ on $y$ is $x = 10.8 + dy$, where $c$ and $d$ are constants. The product moment correlation coefficient is $-0.7214$ and the mean value of $x$ is 7.018.
\begin{enumerate}[label=(\roman*)]
\item Test at the 5% significance level whether there is evidence of non-zero correlation between the variables.
[4]
\item Find the values of $c$ and $d$.
[5]
\item Use an appropriate regression line to estimate the value of $x$ when $y = 3.5$, and comment on the reliability of your estimate.
[2]
\end{enumerate]
\hfill \mbox{\textit{CAIE FP2 2018 Q9 [11]}}