| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Find critical alpha or significance level |
| Difficulty | Challenging +1.2 This is a Further Statistics question requiring calculation of a t-statistic and critical value comparison, plus understanding of CLT robustness. Part (a) involves standard computations (mean, variance, t-value) and consulting tables to find the significance range. Part (b) tests conceptual understanding that large samples (n=128) make t-tests robust to non-normality via CLT. While multi-step, it follows a predictable template with no novel insight required, making it moderately above average difficulty for Further Maths students. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | Mean 30.7 |
| Answer | Marks |
|---|---|
| Range is 0.047 ( 4.7%) | B1 |
| Answer | Marks |
|---|---|
| [5] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2a | Seen or implied |
| Answer | Marks |
|---|---|
| 0.0464 is probably B1M1M0M1A0 | Inequalities can be omitted until final line |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (b) | 128 large enough (for CLT to apply) |
| Hence test not invalidated | M1 |
| Answer | Marks |
|---|---|
| [2] | 2.3 |
| 2.3 | Reason, e.g. sample is large, or n > 25 |
| Answer | Marks |
|---|---|
| Wrong extras, eg “all dists approach normal”: M0 | SC1: CLT applies so not invalidated: B1 |
Question 6:
6 | (a) | Mean 30.7
Biased variance 22.225 (SD 4.714)
22.225 128/127 = 22.4 (SD 4.733)
1 – ((30.7 – 30)/(22.4/128)
Range is 0.047 ( 4.7%) | B1
M1
M1
M1
A1
[5] | 1.1
1.1
1.2
1.1
2.2a | Seen or implied
Calculate variance, allow biased here
128/127 seen, or implied used
Standardise with 128, answer < 0.5 or 50%
Allow > or oe. Awrt 0.047 or 4.7%. CWO.
Wrong or no range: A0. Allow = 4.71 so 5%
0.0464 is probably B1M1M0M1A0 | Inequalities can be omitted until final line
Independent of previous M1.
E.g. 4.6% seen (from no 128/127). Not 0.441
No 128/127, or wrong z: max 3/5
Insuff working: 4.7% 5/5, 4.6% 3/5
0 with some evidence [(22.4)/128]: 4/5
6 | (b) | 128 large enough (for CLT to apply)
Hence test not invalidated | M1
A1
[2] | 2.3
2.3 | Reason, e.g. sample is large, or n > 25
Conclusion, allow “test is valid” or just “no”
Wrong extras, eg “all dists approach normal”: M0 | SC1: CLT applies so not invalidated: B1
SC2: CLT applies as n > (any number other
than 25): max B16 The random variable $X$ was assumed to have a normal distribution with mean $\mu$. Using a random sample of size 128, a significance test was carried out using the following hypotheses.\\
$\mathrm { H } _ { 0 } : \mu = 30$\\
$\mathrm { H } _ { 1 } : \mu > 30$\\
It was found that $\sum x = 3929.6$ and $\sum x ^ { 2 } = 123483.52$. The conclusion of the test was to reject the null hypothesis.
\begin{enumerate}[label=(\alph*)]
\item Determine the range of possible values of the significance level of the test.
\item It was subsequently found that $X$ was not normally distributed.
Explain whether this invalidates the conclusion of the test.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2022 Q6 [7]}}