| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a standard one-sample t-test with straightforward calculations: finding sample mean and standard deviation from given summaries, stating hypotheses, calculating the test statistic, and comparing to critical value. Part (a)(ii) asks for a routine assumption (normality), and part (b) requires stating the switch to a z-test. While it's a Further Maths topic, the execution is mechanical with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \mu = 3\); \(H_1: \mu > 3\) | B1 | States both hypotheses correctly |
| \(\bar{x} = \dfrac{28.8}{9} = 3.2\); \(s^2 = \dfrac{\sum(x-\bar{x})^2}{n-1} = 0.075\) (\(s = 0.27386\)) | B1 | Finds both \(\bar{x}\) and \(s^2\) or \(s\) (condone \(n\) divisor if retrieved in formula) |
| Test statistic \(t = \dfrac{3.2-3}{\sqrt{\dfrac{0.075}{9}}}\) | M1 | Uses formula to find test statistic (PI) |
| Test statistic \(t = 2.19\), \(p = 0.020(4)\) | A1 | States test statistic correctly (AWFW 2.15–2.23) or finds \(p\) |
| Critical value for 1-tail test at 5% level for \(t_8\) distribution is 1.8595; \(2.19 > 1.8595\) or \(0.020(4) < 0.05\) | M1 | Evaluates \(t\) model by comparing test statistic and correct critical value, or by comparing \(p\)-value with 0.05 |
| Reject \(H_0\) | E1 | Infers that \(H_0\) should be rejected |
| As \(2.19 > 1.8595\), the evidence suggests that the impurity, per cent, level in the chemical is too high | E1 | Completes test correctly and gives conclusion in context (conclusion should not be definite) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Assumption: The level of impurity is normally distributed | E1 | Identifies the limitation of the \(t\)-distribution model if data not normally distributed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| It would be a \(z\)-test rather than a \(t\)-test [critical value change to \(z = 1.645\)] | E1 | Makes statement (could be implied by quoting the critical \(z\)-value) that \(z\)-test would be required |
| Would use \(\sigma = 0.25\) not \(\sigma = 0.27386\), or value of \(t_s\) changes to 2.4 | E1 | Mentions change to \(\sigma\) or change to test statistic |
## Question 7(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu = 3$; $H_1: \mu > 3$ | B1 | States both hypotheses correctly |
| $\bar{x} = \dfrac{28.8}{9} = 3.2$; $s^2 = \dfrac{\sum(x-\bar{x})^2}{n-1} = 0.075$ ($s = 0.27386$) | B1 | Finds both $\bar{x}$ and $s^2$ or $s$ (condone $n$ divisor if retrieved in formula) |
| Test statistic $t = \dfrac{3.2-3}{\sqrt{\dfrac{0.075}{9}}}$ | M1 | Uses formula to find test statistic (PI) |
| Test statistic $t = 2.19$, $p = 0.020(4)$ | A1 | States test statistic correctly (AWFW 2.15–2.23) or finds $p$ |
| Critical value for 1-tail test at 5% level for $t_8$ distribution is 1.8595; $2.19 > 1.8595$ or $0.020(4) < 0.05$ | M1 | Evaluates $t$ model by comparing test statistic and correct critical value, or by comparing $p$-value with 0.05 |
| Reject $H_0$ | E1 | Infers that $H_0$ should be rejected |
| As $2.19 > 1.8595$, the evidence suggests that the impurity, per cent, level in the chemical is too high | E1 | Completes test correctly and gives conclusion in context (conclusion should not be definite) |
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## Question 7(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Assumption: The level of impurity is normally distributed | E1 | Identifies the limitation of the $t$-distribution model if data not normally distributed |
---
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| It would be a $z$-test rather than a $t$-test [critical value change to $z = 1.645$] | E1 | Makes statement (could be implied by quoting the critical $z$-value) that $z$-test would be required |
| Would use $\sigma = 0.25$ not $\sigma = 0.27386$, or value of $t_s$ changes to 2.4 | E1 | Mentions change to $\sigma$ or change to test statistic |7 Petroxide Industries produces a chemical used in the production of mobile phone covers for a mobile phone company.
The chemical becomes less effective when the mean level of impurity is greater than 3 per cent.\\
Sunita is the Quality Control manager at Petroxide Industries. After a complaint from the mobile phone company, Sunita obtains a random sample of this chemical from 9 batches.
She measures the level of impurity, $X$ per cent, in each sample.\\
The summarised results are as follows.
$$\sum x = 28.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 0.6$$
7
\begin{enumerate}[label=(\alph*)]
\item (i) Investigate using the $5 \%$ level of significance whether the mean level of impurity in the chemical is greater than 3 per cent.\\[0pt]
[7 marks]\\
7 (a) (ii) State the assumption that it was necessary for you to make in order for the test in part (a)(i) to be valid.\\
7
\item State the changes that would be required to your test in part (a) if you were told that the standard deviation of the level of impurity is known to be 0.25 per cent.\\[0pt]
[2 marks]\\
Turn over for the next question
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics Q7 [10]}}