| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Moderate -0.3 This is a standard two-tailed one-sample t-test with clearly stated hypotheses (μ = 1000 vs μ ≠ 1000). Students must calculate sample mean and standard deviation from given data, compute the t-statistic, and compare to critical values at 5% significance with 11 degrees of freedom. While it requires multiple computational steps, it follows a routine procedure taught extensively in S2 with no conceptual surprises or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| 996 | 1006 | 1009 | 999 | 1007 | 1003 |
| 998 | 1010 | 997 | 996 | 1008 | 1007 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu = 1000\) | B1 | |
| \(H_1: \mu \neq 1000\) | ||
| \[\bar{x} = \frac{12036}{12} = 1003\] | B1 | |
| \(S = 5.444\) | B1 | |
| \(\nu = 12 - 1 = 11\) | B1 | |
| \[t = \frac{\bar{x}-\mu}{S/\sqrt{n}} = \frac{1003-1000}{5.444/\sqrt{12}} = 1.91\] | M1, A1ft, A1 | |
| \(t_{\text{crit}} = \pm 2.201\) | B1√ | |
| Accept \(H_0\) | A1√ | |
| Insufficient evidence to indicate a change in the mean content of sherry in a bottle | E1√ | 10 marks |
$H_0: \mu = 1000$ | B1 | | 2-tailed test
$H_1: \mu \neq 1000$ | |
$$\bar{x} = \frac{12036}{12} = 1003$$ | B1 | |
$S = 5.444$ | B1 | | ($S^2 = 29.6$)
$\nu = 12 - 1 = 11$ | B1 | |
$$t = \frac{\bar{x}-\mu}{S/\sqrt{n}} = \frac{1003-1000}{5.444/\sqrt{12}} = 1.91$$ | M1, A1ft, A1 | |
$t_{\text{crit}} = \pm 2.201$ | B1√ | | (on their ν)
Accept $H_0$ | A1√ | | (on their t-values)
Insufficient evidence to indicate a change in the mean content of sherry in a bottle | E1√ | 10 marks
**Question 8 Total: 10 marks**
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## **TOTAL: 75 marks**8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed.
In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured.
The volumes, in millilitres, of sherry in these bottles are found to be
\begin{center}
\begin{tabular}{ r r r r r r }
996 & 1006 & 1009 & 999 & 1007 & 1003 \\
998 & 1010 & 997 & 996 & 1008 & 1007 \\
\end{tabular}
\end{center}
Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the $5 \%$ level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.
\hfill \mbox{\textit{AQA S2 2006 Q8 [10]}}