| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Standard +0.3 This is a straightforward AS-level statistics question testing standard binomial probability calculations and a one-tailed hypothesis test. Part (a) requires basic binomial probability lookups/calculations, part (b)(i-ii) tests knowledge of sampling methods, and part (b)(iii) is a routine hypothesis test following a standard template. All components are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| 19(a)(i) | Obtains probability from | |
| calculator | AO3.4 | B1 |
| (a)(ii) | Obtains either of these | |
| figures (0.8791, 0.1074) PI | AO3.4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains correct probability | AO1.1b | A1 |
| (b)(i) | Recalls correct name for | |
| sampling method | AO1.2 | B1 |
| (b)(ii) | States that sampling method |
| Answer | Marks | Guidance |
|---|---|---|
| one appropriate weakness | AO3.5b | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| (b)(iii) | States both hypotheses using | |
| correct notation | AO2.5 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| States or uses B(25, 0.2) PI | AO3.3 | M1 |
| Obtains correct probability | AO1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| used instead of 0.109) | AO3.5a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 0 | AO2.2b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| given context | AO3.2a | E1 |
| Total | 11 | |
| TOTAL | 80 |
Question 19:
--- 19(a)(i) ---
19(a)(i) | Obtains probability from
calculator | AO3.4 | B1 | P(X ≤ 2) = 0.678
(a)(ii) | Obtains either of these
figures (0.8791, 0.1074) PI | AO3.4 | M1 | P(X ≤ 3) = 0.8791
P(X = 0) = 0.1074
P(1≤ X ≤ 3) = 0.8791 – 0.1074
= 0.772
Obtains correct probability | AO1.1b | A1
(b)(i) | Recalls correct name for
sampling method | AO1.2 | B1 | Opportunity sampling
(b)(ii) | States that sampling method
is unrepresentative giving
one appropriate weakness | AO3.5b | E1 | The 25 students all come from the
same college and cannot be said to
fairly represent all students. There
could be a regional difference in diet.
(b)(iii) | States both hypotheses using
correct notation | AO2.5 | B1 | H : p = 0.2
0
H : p > 0.2
1
Under H , use X ~ B(25, 0.2)
0
(where X represents number of
students eating 5 or more portions)
P(X ≥ 8) = 0.109
0.109 > 0.05
Hence accept H
0
No significant evidence that more
than 20% eat at least five a day
States or uses B(25, 0.2) PI | AO3.3 | M1
Obtains correct probability | AO1.1b | A1
Evaluates model by
comparing P(X ≥ 8) with
0.05 (condone 0.0468/0.047
used instead of 0.109) | AO3.5a | M1
Infers H accepted
0 | AO2.2b | A1
States correct conclusion in
given context | AO3.2a | E1
Total | 11
TOTAL | 80Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day.
Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size $n$, can be modelled by the binomial distribution B($n$, 0.20).
\begin{enumerate}[label=(\alph*)]
\item There are 10 students in Ellie's statistics class.
Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
\begin{enumerate}[label=(\roman*)]
\item two or fewer eat at least five portions of fruit and vegetables every day; [1 mark]
\item at least one but fewer than four eat at least five portions of fruit and vegetables every day; [2 marks]
\end{enumerate}
\item Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
\begin{enumerate}[label=(\roman*)]
\item Name the sampling method used by Declan. [1 mark]
\item Describe one weakness of this sampling method. [1 mark]
\item Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the 5\% significance level to investigate whether Declan's belief is supported by this evidence. [6 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 Q19 [11]}}