| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | October |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for association |
| Difficulty | Standard +0.3 This is a standard S3 hypothesis testing question covering both Spearman's and Pearson correlation. Part (a) requires routine calculation of rank differences and the Spearman formula. Parts (b) and (d) are textbook hypothesis tests with critical value comparisons. Part (c) uses given summary statistics in a formula. Part (e) tests understanding of when to use each test. All steps are procedural with no novel insight required, making this slightly easier than average. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Price | \(A\) | \(B\) | \(E\) | \(C\) | \(D\) | \(F\) | \(G\) | \(H\) | \(I\) |
| Taste | \(A\) | \(B\) | \(F\) | \(E\) | \(H\) | \(G\) | \(I\) | \(C\) | \(D\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Ranks assigned for Price and Taste across jams A–I | M1 | Attempt to rank each jar for taste and price. At least 4 pairs of ranks correct |
| \(\sum d^2 = [0+0+]16+16+1+9+1+9+4[=56]\) | M1A1 | For an attempt at \(d^2\) row for their ranks (may be implied by \(\sum d^2=56\)); \(\sum d^2=56\) |
| \(r_s = 1 - \frac{6(56)}{9(80)} = \frac{8}{15} = 0.5333\ldots\) | dM1A1 | Dependent on previous M. Using \(1-\frac{6\sum d^2}{9(80)}\); \(\frac{8}{15}\) or awrt 0.533 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho \neq 0\) | B1 | Both hypotheses stated in terms of \(\rho\). Must be two-tail |
| Critical Value \(= 0.7\) | B1 | 0.7 for CV. Allow 0.6 if a one-tail test is used |
| There is no evidence of a relationship between price and taste of strawberry jam | B1ft | For a correct contextualised comment which has price and taste. Follow through their \(r_s\) with their 0.7 (provided \( |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r = \frac{16.4943}{\sqrt{2.0455 \times 243.5556}}\) | M1 | Correct method used |
| \(= 0.7389\ldots\) | A1 | awrt 0.739 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | Both hypotheses stated in terms of \(\rho\). Must be one-tail. If B0 in (b) then allow any letter instead of \(\rho\) consistent with part (b) |
| \(CV = 0.5822\) | B1 | 0.5822. Allow 0.6664 if a two-tail test is used |
| There is evidence of a positive correlation between price and taste of strawberry jam | B1ft | Correct conclusion in context with positive correlation, price and taste. Follow through their 0.5822 and 0.739 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Spearman's rank, as it is unlikely that a joint normal distribution applies, or the marks are a judgement or the marks are not a meaningful scale | B1 | Selecting Spearman's with a suitable reason. Do not allow 'because it is ranked' as a suitable reason |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Ranks assigned for Price and Taste across jams A–I | M1 | Attempt to rank each jar for taste and price. At least 4 pairs of ranks correct |
| $\sum d^2 = [0+0+]16+16+1+9+1+9+4[=56]$ | M1A1 | For an attempt at $d^2$ row for their ranks (may be implied by $\sum d^2=56$); $\sum d^2=56$ |
| $r_s = 1 - \frac{6(56)}{9(80)} = \frac{8}{15} = 0.5333\ldots$ | dM1A1 | Dependent on previous M. Using $1-\frac{6\sum d^2}{9(80)}$; $\frac{8}{15}$ or awrt 0.533 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho \neq 0$ | B1 | Both hypotheses stated in terms of $\rho$. Must be two-tail |
| Critical Value $= 0.7$ | B1 | 0.7 for CV. Allow 0.6 if a one-tail test is used |
| There is no evidence of a relationship between price and taste of strawberry jam | B1ft | For a correct contextualised comment which has price and taste. Follow through their $r_s$ with their 0.7 (provided $|r_s| < 1$) |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r = \frac{16.4943}{\sqrt{2.0455 \times 243.5556}}$ | M1 | Correct method used |
| $= 0.7389\ldots$ | A1 | awrt 0.739 |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | Both hypotheses stated in terms of $\rho$. Must be one-tail. If B0 in (b) then allow any letter instead of $\rho$ consistent with part (b) |
| $CV = 0.5822$ | B1 | 0.5822. Allow 0.6664 if a two-tail test is used |
| There is evidence of a positive correlation between price and taste of strawberry jam | B1ft | Correct conclusion in context with positive correlation, price and taste. Follow through their 0.5822 and 0.739 |
### Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Spearman's rank, as it is unlikely that a joint normal distribution applies, or the marks are a judgement or the marks are not a meaningful scale | B1 | Selecting Spearman's with a suitable reason. Do not allow 'because it is ranked' as a suitable reason |
---3. A cafe owner wishes to know whether the price of strawberry jam is related to the taste of the jam. He finds a website that lists the price per 100 grams and a mark for the taste, out of 100, awarded by a judge, for 9 different strawberry jams $A , B , C , D , E , F , G , H$ and $I$. He then ranks the marks for taste and the prices.
The ranks are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
Rank & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
Price & $A$ & $B$ & $E$ & $C$ & $D$ & $F$ & $G$ & $H$ & $I$ \\
\hline
Taste & $A$ & $B$ & $F$ & $E$ & $H$ & $G$ & $I$ & $C$ & $D$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate Spearman's rank correlation coefficient for these data.
\item Test, at the $5 \%$ level of significance, whether or not there is a relationship between the price and the taste of these strawberry jams. State your hypotheses clearly.
A friend suggests that it would be better to use the price per 100 grams, $c$, and the mark for the taste, $m$, for each strawberry jam rather than rank them.
Given that
$$\mathrm { S } _ { c c } = 2.0455 \quad \mathrm {~S} _ { m m } = 243.5556 \quad \mathrm {~S} _ { c m } = 16.4943$$
\item calculate the product moment correlation coefficient between the price and the mark for taste of these strawberry jams, giving your answer correct to 3 decimal places.
\item Use your value of the product moment correlation coefficient to test, at the $5 \%$ level of significance, whether or not there is evidence of a positive correlation between the price and the mark for taste of these 9 strawberry jams. State your hypotheses clearly.
\item State which of the tests in parts (b) and (d) is more appropriate for the cafe owner to use. Give a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2021 Q3 [14]}}