| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2021 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Multiple binomial probability calculations |
| Difficulty | Moderate -0.8 This is a straightforward application of binomial distribution with standard probability calculations. Part (a) is basic sampling theory recall, parts (b)-(d) involve routine binomial probability computations with clearly stated parameters (n=36, p=0.08, then n=50, p=0.032). No problem-solving insight or multi-step reasoning required—just direct application of formulas or calculator functions. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | Disadvantage: e.g. Not random; cannot use (reliably) for inferences | B1 |
| (b)(i) | Sight or use of \(X \sim B(36, 0.08)\) | M1 |
| \(P(X = 4) = 0.167387...\) | A1 | 1st A1 for awrt 0.167. NB An answer of just awrt 0.167 scores M1(\(\Rightarrow\))1st A1 |
| awrt \(\mathbf{0.167}\) | ||
| (b)(ii) | \([P(X \geq 7)] = 1 - P(X \leq 6) = 0.022233...\) awrt \(\mathbf{0.0222}\) | A1 |
| (c) | \(P(\text{In dance club and dance tango}) = 0.4 \times 0.08 = \mathbf{0.032}\) or \(\frac{4}{125}\) or \(3.2\%\) | B1 |
| (d) | [Let \(T =\) those who can dance the Tango. Sight or use of] \(T \sim B(50, \text{"0.032"})\) | M1 |
| \([P(T < 3) = P(T \leq 2)] = 0.7850815...\) | A1 | A1 for awrt 0.785. Allow MR of 50 (e.g. 30) provided clearly attempting \(P(T \leq 2)\) and score M1A0. |
| awrt \(\mathbf{0.785}\) |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | Disadvantage: e.g. Not random; cannot use (reliably) for inferences | B1 | B1 for a suitable disadvantage. Allow "not random" or "less random" (o.e.). Do NOT allow comments based on time, cost, skew, non-response, or representative. A comment of "negative skew" is B0. |
| (b)(i) | Sight or use of $X \sim B(36, 0.08)$ | M1 | M1 for sight of $B(36, 0.08)$. Allow in words: binomial with $n = 36$ and $p = 0.08$ may be implied by one correct answer to 2sf or sight of $P(X \leq 6) = 0.97776$...i.e. awrt 0.98. Allow for $36 \mathrm{C} 4 \times 0.08^4 \times 0.92^{32}$ as this is "correct use". |
| | $P(X = 4) = 0.167387...$ | A1 | 1st A1 for awrt 0.167. NB An answer of just awrt 0.167 scores M1($\Rightarrow$)1st A1 |
| | awrt $\mathbf{0.167}$ | | |
| (b)(ii) | $[P(X \geq 7)] = 1 - P(X \leq 6) = 0.022233...$ awrt $\mathbf{0.0222}$ | A1 | 2nd A1 for awrt 0.0222 |
| (c) | $P(\text{In dance club and dance tango}) = 0.4 \times 0.08 = \mathbf{0.032}$ or $\frac{4}{125}$ or $3.2\%$ | B1 | B1 for 0.032 o.e. (Can allow for sight of $0.4 \times 0.08$) |
| (d) | [Let $T =$ those who can dance the Tango. Sight or use of] $T \sim B(50, \text{"0.032"})$ | M1 | M1 for sight of $B(50, \text{"0.032"})$ ft their answer to (c) provided it is a probability $\neq 0.08$ may be implied by correct answer. |
| | $[P(T < 3) = P(T \leq 2)] = 0.7850815...$ | A1 | A1 for awrt 0.785. Allow MR of 50 (e.g. 30) provided clearly attempting $P(T \leq 2)$ and score M1A0. |
| | awrt $\mathbf{0.785}$ | | |\begin{enumerate}
\item (a) State one disadvantage of using quota sampling compared with simple random sampling.
\end{enumerate}
In a university 8\% of students are members of the university dance club.\\
A random sample of 36 students is taken from the university.\\
The random variable $X$ represents the number of these students who are members of the dance club.\\
(b) Using a suitable model for $X$, find\\
(i) $\mathrm { P } ( X = 4 )$\\
(ii) $\mathrm { P } ( X \geqslant 7 )$
Only 40\% of the university dance club members can dance the tango.\\
(c) Find the probability that a student is a member of the university dance club and can dance the tango.
A random sample of 50 students is taken from the university.\\
(d) Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango.
\section*{Question 1 continued.}
\hfill \mbox{\textit{Edexcel Paper 3 2021 Q1 [7]}}