| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Moderate -0.3 This is a straightforward one-sample t-test with small sample size (n=8) requiring calculation of sample mean and standard deviation, comparison to critical value, and basic understanding of Type I/II errors. While it involves multiple steps and understanding of hypothesis testing concepts, it follows a standard S2 procedure with no novel problem-solving required—slightly easier than average due to its routine nature and clear structure. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\bar{x} = 225.25\) | B1 | |
| \(s = 5.06 \quad (s^2 = 25.6)\) | B1 | \((\sigma = 4.74)\), \((\sigma^2 = 22.4)\) |
| \(H_0 : \mu = 230\) and \(H_1 : \mu \neq 230\) | B1 | both required |
| \(\nu = 8 - 1 = 7\) | B1 | |
| \(t_{\text{crit}} = \pm 2.365\) | B1 | accept \(t_{\text{crit}} = -2.365\) |
| Test statistic: \(t = \dfrac{225.25 - 230}{5.064/\sqrt{8}} = -2.65\) | M1 | \(\dfrac{225.25 - 230}{4.74/\sqrt{7}} = -2.65\) |
| Correct value \(-2.65\) | A1 | \((-2.66\) to \(-2.65)\) |
| Reject \(H_0\) at 5% level | A1\(\checkmark\) | |
| No evidence to support the producer's claim | E1\(\checkmark\) | 9 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| We have rejected \(H_0\) when in fact \(H_0\) may be true. This indicates that a Type I error may have been made. | B2 | 2 marks total |
## Question 8:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{x} = 225.25$ | B1 | |
| $s = 5.06 \quad (s^2 = 25.6)$ | B1 | $(\sigma = 4.74)$, $(\sigma^2 = 22.4)$ |
| $H_0 : \mu = 230$ and $H_1 : \mu \neq 230$ | B1 | both required |
| $\nu = 8 - 1 = 7$ | B1 | |
| $t_{\text{crit}} = \pm 2.365$ | B1 | accept $t_{\text{crit}} = -2.365$ |
| Test statistic: $t = \dfrac{225.25 - 230}{5.064/\sqrt{8}} = -2.65$ | M1 | $\dfrac{225.25 - 230}{4.74/\sqrt{7}} = -2.65$ |
| Correct value $-2.65$ | A1 | $(-2.66$ to $-2.65)$ |
| Reject $H_0$ at 5% level | A1$\checkmark$ | |
| No evidence to support the producer's claim | E1$\checkmark$ | 9 marks total |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| We have rejected $H_0$ when in fact $H_0$ may be true. This indicates that a Type I error may have been made. | B2 | 2 marks total |8 A jam producer claims that the mean weight of jam in a jar is 230 grams.
\begin{enumerate}[label=(\alph*)]
\item A random sample of 8 jars is selected and the weight of jam in each jar is determined. The results, in grams, are
$$\begin{array} { l l l l l l l l }
220 & 228 & 232 & 219 & 221 & 223 & 230 & 229
\end{array}$$
Assuming that the weight of jam in a jar is normally distributed, test, at the $5 \%$ level of significance, the jam producer's claim.
\item It is later discovered that the mean weight of jam in a jar is indeed 230 grams.
Indicate whether a Type I error, a Type II error or neither has occurred in carrying out the hypothesis test in part (a). Give a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2007 Q8 [11]}}