| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question requiring binomial table lookups and routine application of critical region methodology. While it involves multiple parts and careful probability calculations, it follows textbook procedures without requiring novel insight or complex problem-solving—making it slightly easier than the average A-level question. |
| Spec | 2.05a 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 |
|---|---|---|
| (a) \(\therefore X \sim B(25, 0.20)\) | may be implied | B1; B1 |
| \(P(X \leq 1) = 0.0274\) or \(P(X = 0) = 0.0038\) | need to see at least one. prob for \(X \leq\) no For M1 | M1; A1 |
| \(P(X \leq 9) = 0.9827; \Rightarrow P(X \geq 10) = 0.0173\) | either | A1 |
| \(\therefore CR\) is \(\{X \leq 1 \cup X \geq 10\}\) | A1 | |
| (b) Significance level = 0.0274 + 0.0173 | awrt 0.0447 | B1 |
| \(= 0.0447\) or 4.477% | A1 | |
| (c) \(H_0: p = 0.20\); \(H_1: p < 0.20\) | B1; B1 | |
| Let \(Y\) represent number of bowls with minor defects | B1 | |
| Under \(H_0\) \(Y \sim B(20, 0.20)\) | may be implied | B1 |
| \(P(Y \leq 2)\) or \(P(Y \leq 2) = 0.2061\) or \(P(Y \leq 1) = 0.0692\), CR \(Y \leq 1\) | either | M1; A1 |
| \(= 0.2061\) | A1 | |
| \(0.2061 > 0.10\) or \(0.7939 < 0.9\) or \(> 1\) | their p | M1 |
| Insufficient evidence to suggest that the proportion of defective bowls has decreased. | B1√ |
**(a)** $\therefore X \sim B(25, 0.20)$ | may be implied | B1; B1
$P(X \leq 1) = 0.0274$ or $P(X = 0) = 0.0038$ | need to see at least one. prob for $X \leq$ no For M1 | M1; A1
$P(X \leq 9) = 0.9827; \Rightarrow P(X \geq 10) = 0.0173$ | either | A1
$\therefore CR$ is $\{X \leq 1 \cup X \geq 10\}$ | A1 |
**(b)** Significance level = 0.0274 + 0.0173 | awrt 0.0447 | B1
$= 0.0447$ or 4.477% | A1 |
**(c)** $H_0: p = 0.20$; $H_1: p < 0.20$ | B1; B1
Let $Y$ represent number of bowls with minor defects | B1 |
Under $H_0$ $Y \sim B(20, 0.20)$ | may be implied | B1
$P(Y \leq 2)$ or $P(Y \leq 2) = 0.2061$ or $P(Y \leq 1) = 0.0692$, CR $Y \leq 1$ | either | M1; A1
$= 0.2061$ | A1 |
$0.2061 > 0.10$ or $0.7939 < 0.9$ or $> 1$ | their p | M1
Insufficient evidence to suggest that the proportion of defective bowls has decreased. | B1√ |It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.
\begin{enumerate}[label=(\alph*)]
\item Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5\% as possible. [6]
\item State the actual significance level of the above test. [1]
\end{enumerate}
At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Test, at the 10\% level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2006 Q7 [14]}}