| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2011 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Paired t-test |
| Difficulty | Standard +0.3 This is a straightforward paired t-test application with clear structure: (i) requires recognizing paired data (standard textbook reasoning), (ii) involves routine calculation of differences, mean, standard deviation, and t-statistic with table lookup, (iii) tests a different null hypothesis using the same method. All steps are mechanical and follow standard S3 procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| First test score | 37 | 27 | 38 | 47 | 54 | 27 | 52 | 39 | 62 | 23 |
| Second test score | 47 | 29 | 50 | 44 | 72 | 37 | 63 | 45 | 76 | 32 |
| Answer | Marks | Guidance |
|---|---|---|
| For each student the scores are correlated | B1 | 1 Or equivalent, eg paired |
| Answer | Marks | Guidance |
|---|---|---|
| Increase in score has a normal distribution | B1 | Allow pop of differences~ normal |
| Sample is considered to be a random sample of all students attending the course | B1 | Or equivalent, allow independent |
| \(H_0: \mu_D = 0, H_1: \mu_D > 0\) where \(D=\) increase in scores | B1 | Or \(H_0: \mu_1 \neq \mu_2, H_1: \mu_1 < \mu_2\) not at \(=0\) |
| \(D = 8.9\) | B1 | D may be implied |
| \(s^2 = 35.88\) | B1 | |
| Test statistic \(= 8.9/(\sqrt{10})\) | M1 | Must involve 10 |
| \(= 4.699\) | A1 | Allow ART 4.70 |
| \(v = 9\) CV \(= 3.25\) | B1 | Or \(P(r>4.699)=0.00056<0.005\) |
| \(4.699 > CV\) | M1 | Not OA |
| Reject \(H_0\) and accept that there is sufficient evidence a at the ½ % significance level of an increase in mean scores. | M1 | 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Test statistic \(= (8.9-5)/\sqrt{3.588}=2.059\) | M1A1 | Or \(P(r>2.059)=0.035\) |
| This is significant of an increase at the 5% significance level (CV of 1.833) so director's claim is supported. | A1 | 4(15) Any reasonable significance level with corresponding conclusion; Allow at ½ % |
| Answer | Marks |
|---|---|
| SR: Use of confidence intervals | |
| 99% CI 2-sided \((2.74, 15.1)\) : 99.5% 1-sided \((2.74, \infty)\) 5 is in this interval so not significant at ½ % level A1 OR | M1A1 |
| 90% CI 2-sided \((5.43,12.37)\) ; 95% 1-sided \((5.43, \infty)\) 5 not in this interval so significant at 5% SL | M1A1 |
**7(i)**
For each student the scores are correlated | B1 | 1 Or equivalent, eg paired
**7(ii)**
Increase in score has a normal distribution | B1 | Allow pop of differences~ normal
Sample is considered to be a random sample of all students attending the course | B1 | Or equivalent, allow independent
$H_0: \mu_D = 0, H_1: \mu_D > 0$ where $D=$ increase in scores | B1 | Or $H_0: \mu_1 \neq \mu_2, H_1: \mu_1 < \mu_2$ not at $=0$
$D = 8.9$ | B1 | D may be implied
$s^2 = 35.88$ | B1 |
Test statistic $= 8.9/(\sqrt{10})$ | M1 | Must involve 10
$= 4.699$ | A1 | Allow ART 4.70
$v = 9$ CV $= 3.25$ | B1 | Or $P(r>4.699)=0.00056<0.005$
$4.699 > CV$ | M1 | Not OA
Reject $H_0$ and accept that there is sufficient evidence a at the ½ % significance level of an increase in mean scores. | M1 | 10
SR 2-sample test: (i)B0(ii)B0B1B1M0 Max 2/11
**7(iii)**
Test statistic $= (8.9-5)/\sqrt{3.588}=2.059$ | M1A1 | Or $P(r>2.059)=0.035$
This is significant of an increase at the 5% significance level (CV of 1.833) so director's claim is supported. | A1 | 4(15) Any reasonable significance level with corresponding conclusion; Allow at ½ %
SR 2-sample t-test: $(8.9-5)/s$ M1 Max1/4
SR: Use of confidence intervals | |
99% CI 2-sided $(2.74, 15.1)$ : 99.5% 1-sided $(2.74, \infty)$ 5 is in this interval so not significant at ½ % level A1 OR | M1A1 |
90% CI 2-sided $(5.43,12.37)$ ; 95% 1-sided $(5.43, \infty)$ 5 not in this interval so significant at 5% SL | M1A1 |7 In order to improve their mathematics results 10 students attended an intensive Summer School course. Each student took a test at the start of the course and a similar test at the end of the course. The table shows the scores achieved in each test.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Student & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
First test score & 37 & 27 & 38 & 47 & 54 & 27 & 52 & 39 & 62 & 23 \\
\hline
Second test score & 47 & 29 & 50 & 44 & 72 & 37 & 63 & 45 & 76 & 32 \\
\hline
\end{tabular}
\end{center}
It is desired to test whether there has been an increase in the population mean score.\\
(i) Explain why a two-sample $t$-test would not be appropriate.\\
(ii) Stating any necessary assumptions, carry out a suitable $t$-test at the $\frac { 1 } { 2 } \%$ significance level.\\
(iii) The Summer School director claims that after taking the course the population mean score increases by more than 5 . Is there sufficient evidence for this claim?
\hfill \mbox{\textit{OCR S3 2011 Q7 [15]}}