| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics B AS (Further Statistics B AS) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Confidence interval from software output |
| Difficulty | Moderate -0.8 This question tests basic interpretation of standard statistical software output and recall of confidence interval theory. All parts require straightforward recall: (i) normality assumption, (ii) reading values from output, (iii) defining degrees of freedom as n-1, (iv) standard ways to narrow intervals (larger sample/lower confidence). No calculations, problem-solving, or novel insight required—purely testing whether students understand what the output means. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Result |
| T Estimate of a Mean |
| Mean |
| s |
| SE |
| N |
| df |
| Lower limit |
| Upper limit |
| Interval |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (i) | The underlying parent distribution must be Normal. |
| [1] | Condone just ‘Normal’ | |
| (ii) | 82.4838 < μ < 152.4042 | B1 |
| [1] | Allow to 1 dp 82.5 < μ < 152.4 | |
| (iii) | It means ‘Degrees of freedom’ |
| Answer | Marks |
|---|---|
| values | E1 |
| Answer | Marks |
|---|---|
| [2] | ISW anything after this |
| Answer | Marks |
|---|---|
| test | Must be general, not just 9 – 1 = 8 |
| (iv) | Use a larger sample size |
| Use a lower confidence level | E1 |
| Answer | Marks |
|---|---|
| [2] | Do not allow ‘be less confident’ |
Question 1:
1 | (i) | The underlying parent distribution must be Normal. | E1
[1] | Condone just ‘Normal’
(ii) | 82.4838 < μ < 152.4042 | B1
[1] | Allow to 1 dp 82.5 < μ < 152.4
(iii) | It means ‘Degrees of freedom’
Its value is equal to 1 less than the number of data
values | E1
E1
[2] | ISW anything after this
Only allow n – 1 if n defined
Allow ‘subtracting the number of
dependent variables from n’ (since
they may know about a 2-sample t-
test | Must be general, not just 9 – 1 = 8
(iv) | Use a larger sample size
Use a lower confidence level | E1
E1
[2] | Do not allow ‘be less confident’
Allow ‘Use a smaller confidence
level’1 The birth weights, in kilograms, of a random sample of 9 captive-bred elephants are as follows.
$$\begin{array} { l l l l l l l l l }
94 & 138 & 130 & 118 & 146 & 165 & 82 & 115 & 69
\end{array}$$
A researcher uses software to produce a $99 \%$ confidence interval for the mean birth weight of captive-bred elephants. The output from the software is shown in Fig. 1.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|}
\hline
Result \\
\hline
T Estimate of a Mean \\
\hline
Mean \\
\hline
s \\
\hline
SE \\
\hline
N \\
\hline
df \\
\hline
Lower limit \\
\hline
Upper limit \\
\hline
Interval \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{table}
(i) State an assumption about the distribution of the population from which these weights come that is necessary in order to produce this interval.\\
(ii) State the confidence interval which the software gives, in the form $a < \mu < b$.\\
(iii) Explain
\begin{itemize}
\item what the label df means,
\item how the value of df is calculated for a confidence interval produced using the $t$ distribution.\\
(iv) State two ways in which the researcher could have obtained a narrower confidence interval.
\end{itemize}
\hfill \mbox{\textit{OCR MEI Further Statistics B AS 2018 Q1 [6]}}