| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2022 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a straightforward one-sample t-test question with standard structure: assess normality assumptions, perform the test, and identify an alternative non-parametric test. The calculations are routine (mean, standard deviation, t-statistic), and the conceptual demands (interpreting K-S test p-value, choosing sign test as alternative) are standard Further Statistics content requiring minimal problem-solving insight. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (a) | A Wilcoxon (signed rank) test should be carried out |
| Answer | Marks |
|---|---|
| Normal distribution | B1 |
| Answer | Marks |
|---|---|
| [3] | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 2.2b | B0E0E0 if suggest using a t test even if reasons as below | |
| 11 | (b) | H : population median is 1 |
| Answer | Marks |
|---|---|
| body mass | B1 |
| Answer | Marks |
|---|---|
| [7] | 3.3 |
| Answer | Marks |
|---|---|
| 3.5a | Population median used |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (c) | (Single sample) t test |
| Answer | Marks |
|---|---|
| where µ is population mean increase in body mass | E1 |
| Answer | Marks |
|---|---|
| [3] | 2.2a |
| Answer | Marks |
|---|---|
| 1.1 | All marks are independent |
| Answer | Marks | Guidance |
|---|---|---|
| Value | Val–1 | Abs |
| –0.84 | –1.84 | 1.84 |
| –0.76 | –1.76 | 1.76 |
| –0.16 | –1.16 | 1.16 |
| 0.43 | –0.57 | 0.57 |
| 1.31 | 0.31 | 0.31 |
| 1.32 | 0.32 | 0.32 |
| 1.47 | 0.47 | 0.47 |
| 1.64 | 0.64 | 0.64 |
| 1.93 | 0.93 | 0.93 |
| 2.14 | 1.14 | 1.14 |
Question 11:
11 | (a) | A Wilcoxon (signed rank) test should be carried out
since this test does not require the population to be
Normally distributed,
and the Normal probability plot is not roughly straight
and the p-value is rather low
which both suggest that the data does not come from a
Normal distribution | B1
E1
E1
[3] | 3.3
1.1
2.2b | B0E0E0 if suggest using a t test even if reasons as below
11 | (b) | H : population median is 1
0
H : population median is less than 1
1
Value Val–1 Abs Rank
–0.84 –1.84 1.84 10
–0.76 –1.76 1.76 9
–0.16 –1.16 1.16 8
0.43 –0.57 0.57 4
1.31 0.31 0.31 1
1.32 0.32 0.32 2
1.47 0.47 0.47 3
1.64 0.64 0.64 5
1.93 0.93 0.93 6
2.14 1.14 1.14 7
W = 10 + 9 + 8 + 4 = 31
-
W = 1 + 2 + 3 + 5 + 6 + 7 = 24
+
Test statistic = W = 24 or W = 31
- +
For comparing 24 > 10 or 31 < 45
So do not reject H
0
Insufficient evidence to suggest that the dietary
supplement causes an increase of less than 1 kg in lean
body mass | B1
B1
M1
M1
A1
M1
A1
[7] | 3.3
2.5
1.1
1.1
1.1
3.4
3.5a | Population median used
Both correct Max B1B0 if population not mentioned
NB No marks for test based on Normal distribution
For attempt at ranking
Attempt to calculate either W or W
+ -
Critical value is 10 or 45 Dep on sensible attempt at Wilcoxon,
including finding ranks
Test statistic must be correct
Allow ‘insufficient evidence to suggest that the researcher’s belief
is correct’
11 | (c) | (Single sample) t test
H : µ = 1 kg
0
H : µ < 1 kg
1
where µ is population mean increase in body mass | E1
E1
E1
[3] | 2.2a
1.2
1.1 | All marks are independent
H : Mean increase in body mass in population = 1 kg
0
H : Mean increase in body mass in population < 1 kg
1
Value | Val–1 | Abs | Rank
–0.84 | –1.84 | 1.84 | 10
–0.76 | –1.76 | 1.76 | 9
–0.16 | –1.16 | 1.16 | 8
0.43 | –0.57 | 0.57 | 4
1.31 | 0.31 | 0.31 | 1
1.32 | 0.32 | 0.32 | 2
1.47 | 0.47 | 0.47 | 3
1.64 | 0.64 | 0.64 | 5
1.93 | 0.93 | 0.93 | 6
2.14 | 1.14 | 1.14 | 711 A particular dietary supplement, when taken for a period of 1 month, is claimed to increase lean body mass of adults by an average of 1 kg . A researcher believes that this claim overestimates the increase. She selects a random sample of 10 adults who then each take the supplement for a month. The increases in lean body masses in kg are as follows.
$$\begin{array} { l l l l l l l l l l }
- 0.84 & - 0.76 & - 0.16 & 0.43 & 1.31 & 1.32 & 1.47 & 1.64 & 1.93 & 2.14
\end{array}$$
A Normal probability plot and the $p$-value of the Kolmogorov-Smirnov test for these data are shown below.\\
\includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-09_575_1485_689_242}
\begin{enumerate}[label=(\alph*)]
\item The researcher decides to carry out a hypothesis test in order to investigate the claim.
Comment on the type of hypothesis test that should be used. You should refer to
\begin{itemize}
\item The Normal probability plot
\item The $p$-value of the Kolmogorov-Smirnov test
\item Carry out a test at the $5 \%$ significance level to investigate whether the researcher's belief may be correct.
\item If the Normal probability plot had been different, giving a $p$-value of 0.65 for the KolmogorovSmirnov test, a different procedure could have been used to investigate the researcher's belief.
\item State what alternative test could have been used in this case.
\item State what the hypotheses would have been.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2022 Q11 [13]}}