| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Session | Specimen |
| Marks | 8 |
| 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 application of standard hypothesis testing procedures for a binomial distribution. Part (a) is routine calculator work, part (b) requires setting up H₀: p=0.25 vs H₁: p>0.25 and finding P(X≥16) under B(40,0.25), then comparing to 0.01, and part (c) simply reinterprets the same p-value against 0.05. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 2.04b 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Triangle shape, starting at origin with base on axis, apex between \(t=0\) and \(t=120\) | B1 | 1.1b — Triangle starting at origin with base on \(t\)-axis |
| \(V\) and \(120\) correctly marked on axes | B1 | 1.1b — Allow a delineator |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2} \times 120V = 1500\) | M1 | 3.1b — Identifying correct strategy, equation in \(V\) only |
| \(V = 25\) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Area of triangle = Distance travelled \(= \left(\frac{1}{2} \times 120V\right) = 1500\) | B1 | 2.4 — Area of triangle only depends on base and height |
| This does not depend on \(T\), so \(T\) can take any value where \(0 < T < 120\) | B1 | 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Include a constant speed phase in the motion | B1 | 3.5c — e.g. include a smooth change from acceleration to deceleration phase; variable acceleration and/or deceleration phase |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Triangle shape, starting at origin with base on axis, apex between $t=0$ and $t=120$ | B1 | 1.1b — Triangle starting at origin with base on $t$-axis |
| $V$ and $120$ correctly marked on axes | B1 | 1.1b — Allow a delineator |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2} \times 120V = 1500$ | M1 | 3.1b — Identifying correct strategy, equation in $V$ only |
| $V = 25$ | A1 | 1.1b |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Area of triangle = Distance travelled $= \left(\frac{1}{2} \times 120V\right) = 1500$ | B1 | 2.4 — Area of triangle only depends on base and height |
| This does not depend on $T$, so $T$ can take any value where $0 < T < 120$ | B1 | 2.4 |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Include a constant speed phase in the motion | B1 | 3.5c — e.g. include a smooth change from acceleration to deceleration phase; variable acceleration and/or deceleration phase |
---2. The discrete random variable $X \sim \mathrm {~B} ( 30,0.28 )$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( 5 \leq X < 12 )$.
Past records from a large supermarket show that $25 \%$ of people who buy eggs, buy organic eggs. On one particular day a random sample of 40 people is taken from those that had bought eggs and 16 people are found to have bought organic eggs.
\item Test, at the $1 \%$ significance level, whether or not the proportion $p$ of people who bought organic eggs that day had increased. State your hypotheses clearly.
\item State the conclusion you would have reached if a $5 \%$ significance level had been used for this test.
\section*{(Total for Question 2 is 8 marks)}
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 2 Q2 [8]}}