| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | January |
| Marks | 10 |
| 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 straightforward one-tailed binomial hypothesis test with clearly stated hypotheses, significance level, and sample data. The calculation requires finding P(X ≥ 20) under H₀: p = 0.3, which is routine S2 content. Parts (a)-(c) are basic sampling terminology requiring recall rather than mathematical reasoning. Slightly easier than average due to the direct setup and standard procedure. |
| Spec | 2.01a Population and sample: terminology2.01c Sampling techniques: simple random, opportunity, etc2.01d Select/critique sampling: in context2.05a 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 |
|---|---|---|
| List of all customers (who eat in the restaurant) | B1 | Need idea of list/register/database and 'customer(s)'. Do not allow partial list e.g. 'A list of 50 customers' |
| Answer | Marks | Guidance |
|---|---|---|
| Customer(s) (who ate in the restaurant) | B1 | B1 customer(s) |
| Answer | Marks | Guidance |
|---|---|---|
| Advantage: more/total accuracy, unbiased | B1 | 1st B1 for advantage |
| Disadvantage: time consuming, expensive, difficult to ensure entire population is included | B1 | 2nd B1 for disadvantage |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(X\) = number of customers who would like more choice on the menu | ||
| \(H_0: p = 0.3\), \(H_1: p > 0.3\) | B1 | Need both hypotheses with \(p\) |
| \(X \sim B(50, 0.3)\) | M1 | Using \(B(50, 0.3)\) |
| \(P(X \geq 20) = 1 - P(X \leq 19) = 1 - 0.9152 = 0.0848\) | M1 | For \(1 - P(X \leq 19)\) or \(P(X \leq 20) = 0.9522\) or \(P(X \geq 21) = 0.0478\) leading to critical region \(X > k\) or \(X \geq k\) |
| \(= 0.0848\); or CR: \(P(X \leq 20) = 0.9522\), \(P(X \geq 21) = 0.0478\), \(X \geq 21\) | A1 | awrt 0.0848 or critical region \(X \geq 21\) or \(X > 20\) |
| Do not reject \(H_0\) / not significant / 20 is not in critical region | M1 | Correct conclusion for their probability; may be implied by correct contextual conclusion |
| The percentage of customers who would like more choice on the menu is not more than Bill believes. / There is no evidence to reject Bill's belief. | A1cso | Must mention 'customers' and 'choice' or 'Bill' and 'belief' |
# Question 2:
## Part (a)
| List of all **customers** (who eat in the restaurant) | B1 | Need idea of list/register/database and 'customer(s)'. Do not allow partial list e.g. 'A list of 50 customers' | (1)
## Part (b)
| **Customer(s)** (who ate in the restaurant) | B1 | B1 customer(s) | (1)
## Part (c)
| Advantage: more/total accuracy, unbiased | B1 | 1st B1 for advantage |
| Disadvantage: time consuming, expensive, difficult to ensure entire population is included | B1 | 2nd B1 for disadvantage | (2)
## Part (d)
| Let $X$ = number of customers who would like more choice on the menu | | |
| $H_0: p = 0.3$, $H_1: p > 0.3$ | B1 | Need both hypotheses with $p$ |
| $X \sim B(50, 0.3)$ | M1 | Using $B(50, 0.3)$ |
| $P(X \geq 20) = 1 - P(X \leq 19) = 1 - 0.9152 = 0.0848$ | M1 | For $1 - P(X \leq 19)$ or $P(X \leq 20) = 0.9522$ or $P(X \geq 21) = 0.0478$ leading to critical region $X > k$ or $X \geq k$ |
| $= 0.0848$; or CR: $P(X \leq 20) = 0.9522$, $P(X \geq 21) = 0.0478$, $X \geq 21$ | A1 | awrt 0.0848 or critical region $X \geq 21$ or $X > 20$ |
| Do not reject $H_0$ / not significant / 20 is not in critical region | M1 | Correct conclusion for their probability; may be implied by correct contextual conclusion |
| The percentage of customers who would like more choice on the menu is not more than Bill believes. / There is no evidence to reject Bill's belief. | A1cso | Must mention 'customers' and 'choice' or 'Bill' and 'belief' | (6)2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers' opinions.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable sampling frame for the sample survey.
\item Identify the sampling units.
\item Give one advantage and one disadvantage of taking a census rather than a sample survey.
Bill believes that only $30 \%$ of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
\item Test, at the $5 \%$ significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2014 Q2 [10]}}