| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (lower tail, H₁: p < p₀) |
| Difficulty | Standard +0.3 This is a straightforward S2 hypothesis testing question requiring standard binomial test procedures. Part (a) involves a routine one-tailed test with clear hypotheses and calculation. Parts (b) and (c) require finding critical regions using binomial tables, which is a standard textbook exercise with minimal problem-solving demand. Slightly above average difficulty only due to the two-tailed aspect in part (b), but still very procedural. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.20\), \(H_1: p < 0.20\) | B1, B1 | |
| Let \(X\) represent the number of people buying family size bar. \(X \sim B(30, 0.20)\) | ||
| \(P(X \leq 2) = 0.0442\) | or \(P(X \leq 2) = 0.0442\) awrt 0.044 | M1A1 |
| \(P(X \leq 3) = 0.1227\) CR \(X \leq 2\) | ||
| 0.0442 < 5%, so significant. | Significant | M1 |
| There is evidence that the no. of family size bars sold is lower than usual. | A1 (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.02\), \(H_1: p \neq 0.02\) | B1 | |
| \(\lambda = 4\) etc ok both | ||
| Let \(Y\) represent the number of gigantic bars sold. | ||
| \(Y \sim B(200, 0.02) \Rightarrow Y \sim \text{Po}(4)\) | can be implied below | M1 |
| \(P(Y = 0) = \mathbf{0.0183}\) and \(P(Y \leq 8) = 0.9786 \Rightarrow P(Y \geq 9) = \mathbf{0.0214}\) | first, either | B1, B1 |
| Critical region \(Y = 0 \cup Y \geq 9\) | \(Y \leq 0\) ok | B1, B1 |
| N.B. Accept exact Bin: 0.0176 and 0.0202 |
| Answer | Marks | Guidance |
|---|---|---|
| Significance level \(= 0.0183 + 0.0214 = 0.0397\) | awrt 0.04 | B1 (1 mark) |
**Part (a)**
$H_0: p = 0.20$, $H_1: p < 0.20$ | | B1, B1 |
Let $X$ represent the number of people buying family size bar. $X \sim B(30, 0.20)$ | |
$P(X \leq 2) = 0.0442$ | or $P(X \leq 2) = 0.0442$ awrt 0.044 | M1A1 |
$P(X \leq 3) = 0.1227$ CR $X \leq 2$ | | |
0.0442 < 5%, so significant. | Significant | M1 |
There is evidence that the no. of family size bars sold is lower than usual. | | A1 (6 marks) |
**Part (b)**
$H_0: p = 0.02$, $H_1: p \neq 0.02$ | | B1 |
$\lambda = 4$ etc ok both | |
Let $Y$ represent the number of gigantic bars sold. | |
$Y \sim B(200, 0.02) \Rightarrow Y \sim \text{Po}(4)$ | can be implied below | M1 |
$P(Y = 0) = \mathbf{0.0183}$ and $P(Y \leq 8) = 0.9786 \Rightarrow P(Y \geq 9) = \mathbf{0.0214}$ | first, either | B1, B1 |
Critical region $Y = 0 \cup Y \geq 9$ | $Y \leq 0$ ok | B1, B1 |
N.B. Accept exact Bin: 0.0176 and 0.0202 | | |
**Part (c)**
Significance level $= 0.0183 + 0.0214 = 0.0397$ | awrt 0.04 | B1 (1 mark) |
---6. Past records from a large supermarket show that $20 \%$ of people who buy chocolate bars buy the family size bar. On one particular day a random sample of 30 people was taken from those that had bought chocolate bars and 2 of them were found to have bought a family size bar.
\begin{enumerate}[label=(\alph*)]
\item Test at the $5 \%$ significance level, whether or not the proportion $p$, of people who bought a family size bar of chocolate that day had decreased. State your hypotheses clearly.
The manager of the supermarket thinks that the probability of a person buying a gigantic chocolate bar is only 0.02 . To test whether this hypothesis is true the manager decides to take a random sample of 200 people who bought chocolate bars.
\item Find the critical region that would enable the manager to test whether or not there is evidence that the probability is different from 0.02 . The probability of each tail should be as close to $2.5 \%$ as possible.
\item Write down the significance level of this test.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2007 Q6 [13]}}