| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Standard +0.3 This is a standard A-level S2 hypothesis testing question requiring routine application of t-test procedures with small sample size. While it involves multiple parts including confidence intervals, hypothesis testing, and error types, all steps follow textbook methods with no novel insight required. The calculations are straightforward (finding mean, standard deviation, applying t-distribution), making it slightly easier than average for A-level statistics. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\bar{x} = 43.5\) | B1 | |
| \(s = 2\ (s^2 = 4)\) | B1 | |
| Assumption: Weights of boxes are normally distributed | B1 | |
| \(t_{0.975} = 2.365\) | B1 | |
| \(43.5 \pm 2.365 \times \frac{2}{\sqrt{8}}\) | M1 | |
| \(43.5 \pm 1.6723\) | ||
| \(\Rightarrow (41.8, 45.2)\) | A1 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| CI contains mean (45), Bishen's belief probably justified | B1 dep | Must be clear use of 45 and not 43.5 |
| or [Since 45 within CI] but close to upper limit, there is some evidence that Bishen's belief is untrue [but the evidence is not significant at 5%.] (75% of sample less than 45 grams) | (B1) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \mu = 45\); \(H_1: \mu < 45\) | B1 | (both) |
| Test statistic: \(t = \frac{43.5 - 45}{\frac{2}{\sqrt{8}}}\) | M1 | |
| \(= -2.12\) | A1 | \(P(t_7 < -2.12) = 0.035791\) |
| \(\nu = 7 \Rightarrow t_{crit} = -1.895\) | B1 | \(< 0.05\) |
| \(\Rightarrow\) Reject \(H_0\) | A1 | |
| Evidence at the 5% level of significance to support Abi's claim that mean content \(< 45\) grams | E1 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Type I error | B1 | |
| have/may have rejected \(H_0\) when \(H_0\) true | B1 | 2 |
| or No error | (B1) | |
| have/may have accepted \(H_0\) when \(H_0\) true | (B1) | Clear statement |
| TOTAL | 75 |
# Question 6(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\bar{x} = 43.5$ | B1 | |
| $s = 2\ (s^2 = 4)$ | B1 | |
| Assumption: Weights of boxes are normally distributed | B1 | |
| $t_{0.975} = 2.365$ | B1 | |
| $43.5 \pm 2.365 \times \frac{2}{\sqrt{8}}$ | M1 | |
| $43.5 \pm 1.6723$ | | |
| $\Rightarrow (41.8, 45.2)$ | A1 | 6 | (AWRT) |
---
# Question 6(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| CI contains mean (45), Bishen's belief probably justified | B1 dep | Must be clear use of 45 and **not** 43.5 |
| **or** [Since 45 within CI] but close to upper limit, there is some evidence that Bishen's belief is untrue [but the evidence is not significant at 5%.] (75% of sample less than 45 grams) | (B1) | 2 |
---
# Question 6(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu = 45$; $H_1: \mu < 45$ | B1 | (both) |
| Test statistic: $t = \frac{43.5 - 45}{\frac{2}{\sqrt{8}}}$ | M1 | |
| $= -2.12$ | A1 | $P(t_7 < -2.12) = 0.035791$ |
| $\nu = 7 \Rightarrow t_{crit} = -1.895$ | B1 | $< 0.05$ |
| $\Rightarrow$ Reject $H_0$ | A1 | |
| Evidence at the 5% level of significance to support Abi's claim that **mean** content $< 45$ grams | E1 | 6 |
---
# Question 6(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Type I error | B1 | |
| have/may have rejected $H_0$ when $H_0$ true | B1 | 2 | Clear statement |
| **or** No error | (B1) | |
| have/may have accepted $H_0$ when $H_0$ true | (B1) | Clear statement |
| **TOTAL** | **75** | |6 Bishen believes that the mean weight of boxes of black peppercorns is 45 grams. Abi, thinking that this is not the case, weighs, in grams, a random sample of 8 boxes of black peppercorns, with the following results.
$$\begin{array} { l l l l l l l l }
44 & 44 & 43 & 46 & 42 & 40 & 43 & 46
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Construct a $95 \%$ confidence interval for the mean weight of boxes of black peppercorns, stating any assumption that you make.
\item Comment on Bishen's belief.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Abi claims that the mean weight of boxes of black peppercorns is less than 45 grams. Test this claim at the $5 \%$ level of significance.
\item If Bishen's belief is true, state, with a reason, what type of error, if any, may have occurred when conclusions to the test in part (b)(i) were drawn.\\
(2 marks)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2009 Q6 [16]}}