| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2009 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Paired sample t-test |
| Difficulty | Standard +0.3 This is a standard paired t-test application with straightforward calculations. Part (i) requires routine execution of a hypothesis test procedure (calculate differences, find mean and standard deviation, compute t-statistic, compare to critical value). Part (ii) adds a mild algebraic twist but follows directly from understanding how adding a constant affects the test statistic. The question is slightly easier than average because it's a textbook application with clear structure and no conceptual surprises, though it does require careful arithmetic and understanding of the paired t-test framework. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| First test | 38 | 27 | 55 | 43 | 32 | 24 | 51 | 46 |
| Second test | 37 | 26 | 57 | 43 | 30 | 26 | 54 | 48 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Requires population of (2nd mark – 1st mark) to be normally distributed | B1 | |
| \(H_0: \mu_d = 0\), \(H_1: \mu_d > 0\) | M1 | |
| \(T_2 - T_1\): \(-1, -1, 2, 0, -2, 2, 3, 2\) | B1B1 | |
| \(\bar{d} = 0.625\), \(s^2 = 3.411\) (\(3^{23}/_{56}\) or \(^{191}/_{56}\)) | B1 | |
| Use 2.998 | M1 | |
| EITHER: \(t = 0.625/\sqrt{3.411/8} = 0.957\) | A1 | M0 if clearly \(z\) |
| OR: CV(CR), \(\bar{d} \geq 2.998\sqrt{3.411/8} = 1.958\) | M1, A1 | |
| EITHER \(0.957 < 2.998\) OR \(0.625 < 1.958\) | M1 | |
| Do not reject \(H_0\), there is insufficient evidence of improvement | 8 | With comparison and conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(E(X_2 - X_1 + k) = 0.625 + k\) | M1 | |
| Requires \((0.625+k)/\sqrt{3.411/8} \geq 2.998\) | \(\text{A1}\sqrt{}\) | |
| Giving \(k \geq 1.33\) | A1 3 | Allow 1.33 |
| Increase each mark by 2 |
# Question 7:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Requires population of (2nd mark – 1st mark) to be normally distributed | B1 | |
| $H_0: \mu_d = 0$, $H_1: \mu_d > 0$ | M1 | |
| $T_2 - T_1$: $-1, -1, 2, 0, -2, 2, 3, 2$ | B1B1 | |
| $\bar{d} = 0.625$, $s^2 = 3.411$ ($3^{23}/_{56}$ or $^{191}/_{56}$) | B1 | |
| Use 2.998 | M1 | |
| EITHER: $t = 0.625/\sqrt{3.411/8} = 0.957$ | A1 | M0 if clearly $z$ |
| OR: CV(CR), $\bar{d} \geq 2.998\sqrt{3.411/8} = 1.958$ | M1, A1 | |
| EITHER $0.957 < 2.998$ OR $0.625 < 1.958$ | M1 | |
| Do not reject $H_0$, there is insufficient evidence of improvement | **8** | With comparison and conclusion |
## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $E(X_2 - X_1 + k) = 0.625 + k$ | M1 | |
| Requires $(0.625+k)/\sqrt{3.411/8} \geq 2.998$ | $\text{A1}\sqrt{}$ | |
| Giving $k \geq 1.33$ | A1 **3** | Allow 1.33 |
| Increase each mark by 2 | | |
---7 A tutor gives a randomly selected group of 8 students an English Literature test, and after a term's further teaching, she gives the group a similar test. The marks for the two tests are given in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Student & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ \\
\hline
First test & 38 & 27 & 55 & 43 & 32 & 24 & 51 & 46 \\
\hline
Second test & 37 & 26 & 57 & 43 & 30 & 26 & 54 & 48 \\
\hline
\end{tabular}
\end{center}
(i) Stating a necessary condition, show by carrying out a suitable $t$-test, at the $1 \%$ significance level, that the marks do not give evidence of an improvement.\\
(ii) The tutor later found that she had marked the second test too severely, and she decided to add a constant amount $k$ to each mark. Find the least integer value of $k$ for which the increased marks would give evidence of improvement at the $1 \%$ significance level.
\hfill \mbox{\textit{OCR S3 2009 Q7 [11]}}