| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2017 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Paired sample t-test |
| Difficulty | Standard +0.3 This is a straightforward application of a paired t-test with clear structure: calculate differences, find mean and standard deviation, perform the test, and interpret. The conceptual parts (justifying paired test, stating assumptions, Type II error) are standard bookwork. While S4 is Further Maths content, this question requires only routine execution of a standard procedure with no novel insight or complex reasoning. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Player | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| First round | 76 | 80 | 72 | 78 | 83 | 88 | 81 | 72 |
| Final round | 70 | 78 | 75 | 75 | 79 | 84 | 83 | 69 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The data is collected in pairs or samples not independent | B1 | Allow: same person has been used; do not allow "2 data sets" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The differences are normally distributed | B1 | Must mention "differences" and "normal" distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(d\): 6, 2, −3, 3, 4, 4, −2, 3 | M1 | For attempting the \(d\)s; at least 2 correct implied by figures |
| \(\Sigma d = 17\), \(\Sigma d^2 = 103\), \(\bar{d} = \pm 2.125\), \(s_d = 3.09\ldots\) (Var \(= 9.55\ldots\)) | M1 M1 | M1 for attempting \(\bar{d}\); M1 for \(s_d\) or \(s_d^2\) |
| \(H_0: \mu_d = 1\), \(H_1: \mu_d > 1\) | B1 | Both hypotheses correct in terms of \(\mu\) or \(\mu_d\); must match their differences |
| \(t = \pm\left(\dfrac{2.125 - 1}{3.09/\sqrt{8}}\right) = \pm 1.02947\ldots\) | M1A1 | M1 for correct test statistic \(\dfrac{\bar{d}-1}{s_d/\sqrt{8}}\); A1 awrt 1.03 |
| Critical value \(t_7(5\%) = \pm 1.895\) (1 tail) | B1 | awrt 1.895; sign must match their \(t\)-value |
| Not significant. Insufficient evidence to support that the score in the final round is more than 1 below the score in the first round, or insufficient evidence to support the coach's belief | A1ft | ft \(t\)-value if awarded both B marks; correct comment in context – bold words needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The idea that "the coach's belief is rejected when it is in fact true" | B1 | B1 for \(H_1\) is rejected when it is in fact true |
| Correct contextual statement | B1 | B2 correct contextual statement |
# Question 4:
## Part (a)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| The data is collected in pairs or samples not independent | B1 | Allow: same person has been used; do not allow "2 data sets" |
## Part (a)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| The differences are normally distributed | B1 | Must mention "differences" and "normal" distribution |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $d$: 6, 2, −3, 3, 4, 4, −2, 3 | M1 | For attempting the $d$s; at least 2 correct implied by figures |
| $\Sigma d = 17$, $\Sigma d^2 = 103$, $\bar{d} = \pm 2.125$, $s_d = 3.09\ldots$ (Var $= 9.55\ldots$) | M1 M1 | M1 for attempting $\bar{d}$; M1 for $s_d$ or $s_d^2$ |
| $H_0: \mu_d = 1$, $H_1: \mu_d > 1$ | B1 | Both hypotheses correct in terms of $\mu$ or $\mu_d$; must match their differences |
| $t = \pm\left(\dfrac{2.125 - 1}{3.09/\sqrt{8}}\right) = \pm 1.02947\ldots$ | M1A1 | M1 for correct test statistic $\dfrac{\bar{d}-1}{s_d/\sqrt{8}}$; A1 awrt 1.03 |
| Critical value $t_7(5\%) = \pm 1.895$ (1 tail) | B1 | awrt 1.895; sign must match their $t$-value |
| Not significant. Insufficient evidence to support that the **score** in the **final** round is **more than 1 below** the score in the **first** round, or insufficient evidence to support the **coach's** belief | A1ft | ft $t$-value if awarded both B marks; correct comment in context – bold words needed |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| The idea that "the coach's belief is rejected when it is in fact true" | B1 | B1 for $H_1$ is rejected when it is in fact true |
| Correct contextual statement | B1 | B2 correct contextual statement |
---4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Player & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ \\
\hline
First round & 76 & 80 & 72 & 78 & 83 & 88 & 81 & 72 \\
\hline
Final round & 70 & 78 & 75 & 75 & 79 & 84 & 83 & 69 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item State why a paired $t$-test is suitable for use with these data.
\item State an assumption that needs to be made in order to carry out a paired $t$-test in this case.
\end{enumerate}\item Test, at the $5 \%$ level of significance, whether or not there is evidence to support the coach's belief. Show your working clearly.
\item Explain, in the context of the coach's belief, what a Type II error would be in this case.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2017 Q4 [12]}}