| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2014 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | One-tailed test for positive correlation |
| Difficulty | Standard +0.3 This is a standard S3 hypothesis testing question requiring routine application of correlation coefficient formulas and critical value comparison. Parts (a)-(b) involve straightforward calculation and one-tailed test setup with given summary statistics. Parts (c)-(d) require ranking data and applying Spearman's formula, then another standard hypothesis test. All procedures are textbook exercises with no novel problem-solving required, though the multi-part structure and need for critical value tables places it slightly above average difficulty. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Man | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) |
| Height \(( x )\) | 1.68 | 1.74 | 1.75 | 1.76 | 1.78 | 1.82 | 1.84 | 1.88 | 1.98 |
| Weight \(( y )\) | 75 | 76 | 100 | 77 | 90 | 95 | 110 | 96 | 120 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r = \dfrac{9.3433}{\sqrt{0.0632 \times 1957.5556}}\) | M1 | Clear use of \(r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\) |
| \(= 0.840\) | A1 | A1 0.840 cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | 1st B1 both hypotheses in terms of \(\rho\), one tail \(H_1\) must be compatible with their \(r\). Hypotheses just in words e.g. "no correlation" score B0 |
| Critical value \(= 0.5822\) | B1 | 2nd B1 for 0.5822 cao |
| \(0.840 > 0.5822\), there is evidence to reject \(H_0\) | M1 | M1 for statement comparing 'their \(r\)' with 'their cv' |
| There is evidence of a positive correlation between a man's height and his weight | A1ft | A1 for correct contextualised comment. Must mention positive correlation, 1-tailed test, height and weight. Follow through their \(r\) and cv (provided \( |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Actual weight rankings correct | B1 | 1st B1 for attempt to rank actual weight / Peter's order with at least 4 correct |
| Peter's order rankings correct | B1 | 2nd B1 for correct rankings for both (one or both may be reversed) |
| \(\sum d^2 = 70\) | M1A1 | 1st M1 for use of \(\sum d^2\) with at least 4 values correct and attempt to add. 1st A1 for 70 or 170 with reversed rankings |
| \(r_s = 1 - \dfrac{6\sum d^2}{n(n^2-1)} = 1 - \dfrac{6 \times 70}{9(81-1)}\) | dM1 | 2nd dM1 for use of correct formula, follow through their \(\sum d^2\). Dependent on 1st M1. If answer not correct, correct expression required |
| \(= 0.417\) | A1 | 2nd A1 for awrt 0.417 or \(\dfrac{5}{12}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | 1st B1 both hypotheses in terms of \(\rho\) or \(\rho_s\). One tail \(H_1\) must be compatible with their ranking. Hypotheses just in words score B0 |
| Critical value \(0.600\) | B1 | 2nd B1 for cv of 0.6(00) cao. Their cv must be compatible with \(H_1\) |
| \((0.417 < 0.600)\) There is insufficient evidence to reject \(H_0\) | M1 | M1 for statement comparing 'their \(r\)' with 'their cv' |
| Peter does not have the ability to correctly order men, by weight, from their photograph | A1 | A1 for correct contextualised comment. Must mention Peter and Men. Follow through their \(r\) and cv (provided \( |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r = \dfrac{9.3433}{\sqrt{0.0632 \times 1957.5556}}$ | M1 | Clear use of $r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ |
| $= 0.840$ | A1 | A1 0.840 cao |
**(2 marks)**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | 1st B1 both hypotheses in terms of $\rho$, one tail $H_1$ must be compatible with their $r$. Hypotheses just in words e.g. "no correlation" score B0 |
| Critical value $= 0.5822$ | B1 | 2nd B1 for 0.5822 cao |
| $0.840 > 0.5822$, there is evidence to reject $H_0$ | M1 | M1 for statement comparing 'their $r$' with 'their cv' |
| There is evidence of a positive correlation between a man's height and his weight | A1ft | A1 for correct contextualised comment. Must mention positive correlation, 1-tailed test, height and weight. Follow through their $r$ and cv (provided $|cv| < 1$ and $|r| < 1$) |
**(4 marks)**
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Actual weight rankings correct | B1 | 1st B1 for attempt to rank actual weight / Peter's order with at least 4 correct |
| Peter's order rankings correct | B1 | 2nd B1 for correct rankings for both (one or both may be reversed) |
| $\sum d^2 = 70$ | M1A1 | 1st M1 for use of $\sum d^2$ with at least 4 values correct and attempt to add. 1st A1 for 70 or 170 with reversed rankings |
| $r_s = 1 - \dfrac{6\sum d^2}{n(n^2-1)} = 1 - \dfrac{6 \times 70}{9(81-1)}$ | dM1 | 2nd dM1 for use of correct formula, follow through their $\sum d^2$. Dependent on 1st M1. If answer not correct, correct expression required |
| $= 0.417$ | A1 | 2nd A1 for awrt 0.417 or $\dfrac{5}{12}$ |
**(6 marks)**
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | 1st B1 both hypotheses in terms of $\rho$ or $\rho_s$. One tail $H_1$ must be compatible with their ranking. Hypotheses just in words score B0 |
| Critical value $0.600$ | B1 | 2nd B1 for cv of 0.6(00) cao. Their cv must be compatible with $H_1$ |
| $(0.417 < 0.600)$ There is insufficient evidence to reject $H_0$ | M1 | M1 for statement comparing 'their $r$' with 'their cv' |
| Peter does not have the ability to correctly order men, by weight, from their photograph | A1 | A1 for correct contextualised comment. Must mention Peter and Men. Follow through their $r$ and cv (provided $|cv| < 1$ and $|r_s| < 1$) |
**(4 marks) — Total 16**8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | }
\hline
Man & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ \\
\hline
Height $( x )$ & 1.68 & 1.74 & 1.75 & 1.76 & 1.78 & 1.82 & 1.84 & 1.88 & 1.98 \\
\hline
Weight $( y )$ & 75 & 76 & 100 & 77 & 90 & 95 & 110 & 96 & 120 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { S } _ { x x } = 0.0632 , \mathrm {~S} _ { y y } = 1957.5556$ and $\mathrm { S } _ { x y } = 9.3433$ calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men.
\item Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a $5 \%$ significance level. State your hypotheses clearly.
Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order
$$A C E B G D I F H$$
\item Find, to 3 decimal places, Spearman's rank correlation coefficient between Peter's order and the actual order.
\item Use your value of Spearman's rank correlation coefficient to test for evidence of Peter's ability to correctly order men, by their weight, from their photographs. Use a 5\% significance level and state your hypotheses clearly.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2014 Q8 [16]}}