| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| 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 with clear hypotheses (one-tailed test at 1% level). Students must calculate sample mean and standard deviation from given data, then apply the standard t-test procedure. While it requires multiple computational steps and understanding of hypothesis testing framework, it's a routine application of a standard technique with no conceptual surprises—slightly easier than average due to small sample size and clear setup. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \mu = 30\), \(H_1: \mu > 30\) | B1 | |
| \(\bar{x} = 33.5\), \(s = 4.25\) (\(s^2 = 18.06\)) | B1B1 | \(\sigma = 4.03\) (\(\sigma^2 = 16.25\)) |
| Under \(H_0\), \(\bar{X} \sim N\!\left(30, \dfrac{4.25^2}{10}\right)\) | \(\downarrow\) | |
| \(t = \dfrac{33.5 - 30}{4.25/\sqrt{10}} = 2.60\) | M1A1 | \(\dfrac{33.5-30}{4.03/\sqrt{9}}\) \((2.6\)–\(2.61)\) |
| \(t_{\text{crit}} = 2.821\) | B1 | |
| Do not reject \(H_0\) | ||
| Insufficient evidence at the 1% level of significance that Jasmine's teacher is underestimating the time that it takes to complete the homework assignments. | E1\(\checkmark\) | Total: 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Times are Normally distributed | B1 | Total: 1 |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \mu = 30$, $H_1: \mu > 30$ | B1 | |
| $\bar{x} = 33.5$, $s = 4.25$ ($s^2 = 18.06$) | B1B1 | $\sigma = 4.03$ ($\sigma^2 = 16.25$) |
| Under $H_0$, $\bar{X} \sim N\!\left(30, \dfrac{4.25^2}{10}\right)$ | | $\downarrow$ |
| $t = \dfrac{33.5 - 30}{4.25/\sqrt{10}} = 2.60$ | M1A1 | $\dfrac{33.5-30}{4.03/\sqrt{9}}$ $(2.6$–$2.61)$ |
| $t_{\text{crit}} = 2.821$ | B1 | |
| Do not reject $H_0$ | | |
| Insufficient evidence at the 1% level of significance that Jasmine's teacher is underestimating the time that it takes to complete the homework assignments. | E1$\checkmark$ | **Total: 7** |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Times are Normally distributed | B1 | **Total: 1** |
---5 Jasmine's French teacher states that a homework assignment should take, on average, 30 minutes to complete.
Jasmine believes that he is understating the mean time that the assignment takes to complete and so decides to investigate. She records the times, in minutes, that it takes for a random sample of 10 students to complete the French assignment, with the following results:
$$\begin{array} { l l l l l l l l l l }
29 & 33 & 36 & 42 & 30 & 28 & 31 & 34 & 37 & 35
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Test, at the $1 \%$ level of significance, Jasmine's belief that her French teacher has understated the mean time that it should take to complete the homework assignment.
\item State an assumption that you must make in order for the test used in part (a) to be valid.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2007 Q5 [8]}}