| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Moderate -0.3 This is a standard one-tailed binomial hypothesis test with straightforward setup (H₀: p=0.14, H₁: p>0.14, n=60, x=15). Part (a) tests understanding of one-tailed vs two-tailed tests, part (b) requires routine calculation of P(X≥15) and comparison to 5% significance level, and part (c) asks for a standard assumption (random sample). The calculations are accessible and the question follows a textbook template, making it slightly easier than average for AS-level statistics. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| If the campaign is effective, then the proportion of under 30s visitors will be greater than 14%. So, a one-tailed test is required. | E1 | Explains why this is a one-tailed test. Must be in context. Condone explanation using 'number of' linked to increase |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_1: p > 0.14\) | B1 | States both hypotheses correctly for a one-tailed test. Accept population proportion for \(p\). Accept 14%, but not \(x=\) or \(\bar{x}=\) or \(\mu=\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \geq 15) = 1 - P(X \leq 14) = 1 - 0.98351\ldots = 0.01649\) | M1 | States model used. PI by 0.016(5), 0.0071(5), 0.035, 0.0029, 0.0093 |
| Answer | Marks | Guidance |
|---|---|---|
| As \(0.0165 < 0.05\) | A1 | Evaluates using calculator \(= 0.016(5)\). Condone \(0.0071(5)\) for A1 |
| Reject \(H_0\) | A1 | Compares \(0.016(5)\) to \(0.05\) and rejects \(H_0\). No ft here. Must see clear comparison (inequality or diagram) |
| There is sufficient evidence to suggest that the advertising campaign has been effective. | R1 | Concludes correctly in context. CSO 'sufficient evidence' OE required. Only award for full complete correct solution. |
| Answer | Marks | Guidance |
|---|---|---|
| The sample would need to be a Random sample | E1 | Recalls that the sample would need to be random. Accept 'not biased' OE |
# Question 18(a):
If the campaign is effective, then the proportion of under 30s visitors will be greater than 14%. So, a one-tailed test is required. | E1 | Explains why this is a one-tailed test. Must be in context. Condone explanation using 'number of' linked to increase
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# Question 18(b):
$X$ is 'No of under 30's visitors to the website'
$H_0: p = 0.14$
$H_1: p > 0.14$ | B1 | States both hypotheses correctly for a one-tailed test. Accept population proportion for $p$. Accept 14%, but not $x=$ or $\bar{x}=$ or $\mu=$
Under $H_0$: $X \sim B(60, 0.14)$
$P(X \geq 15) = 1 - P(X \leq 14) = 1 - 0.98351\ldots = 0.01649$ | M1 | States model used. PI by 0.016(5), 0.0071(5), 0.035, 0.0029, 0.0093
$= 0.0165$
As $0.0165 < 0.05$ | A1 | Evaluates using calculator $= 0.016(5)$. Condone $0.0071(5)$ for A1
Reject $H_0$ | A1 | Compares $0.016(5)$ to $0.05$ and rejects $H_0$. No ft here. Must see clear comparison (inequality or diagram)
There is sufficient evidence to suggest that the advertising campaign has been effective. | R1 | Concludes correctly in context. CSO 'sufficient evidence' OE required. Only award for full complete correct solution.
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# Question 18(c):
The sample would need to be a Random sample | E1 | Recalls that the sample would need to be random. Accept 'not biased' OE18 It is known from previous data that 14\% of the visitors to a particular cookery website are under 30 years of age.
To encourage more visitors under 30 years of age a large advertising campaign took place to target this age group.
To test whether the campaign was effective, a sample of 60 visitors to the website was selected. It was found that 15 of the visitors were under 30 years of age.
18
\begin{enumerate}[label=(\alph*)]
\item Explain why a one-tailed hypothesis test should be used to decide whether the sample provides evidence that the campaign was effective.
18
\item Carry out the hypothesis test at the $5 \%$ level of significance to investigate whether the sample provides evidence that the proportion of visitors under 30 years of age has increased.\\
18
\item State one necessary assumption about the sample for the distribution used in part (b) to be valid.\\[0pt]
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2021 Q18 [7]}}