| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Moderate -0.3 This is a standard S2 hypothesis testing question covering routine binomial test procedures. Parts (a)-(d) involve textbook applications of critical regions and two-tailed tests, while part (e) is a one-tailed test. All steps follow established algorithms with no novel problem-solving required, though the multi-part structure and cumulative probability calculations place it slightly below average difficulty for A-level. |
| Spec | 2.04b Binomial distribution: as model B(n,p)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) 2 outcomes/faulty or not faulty/success or fail | B1; B1 | (2) |
| A constant probability | ||
| Independence | ||
| Fixed number of trials (fixed \(n\)) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \geq 19) = 0.0287\) | M1; A1 A1 | (3) |
| CR \(X \leq 6\) and \(X \geq 19\) | ||
| (c) \(0.0194 + 0.0287 = 0.0481\) | M1A1 | (2) |
| (d) 8(It) is not in the Critical region or 8(It) is not significant or 0.0916 > 0.025; There is evidence that the probability of a faulty bolt is 0.25 or the company's claim is correct. | M1; A1ft | (2) |
| (e) \(H_0: p = 0.25\) \(H_1: p < 0.25\) | B1B1; M1A1 | (6) |
**(a)** 2 outcomes/faulty or not faulty/success or fail | B1; B1 | (2) | B1 B1 one mark for each of any of the four statements. Give first B1 if only one correct statement given. No context needed.
A constant probability | | |
Independence | | |
Fixed number of trials (fixed $n$) | | |
**(b)** $X \sim B(50, 0.25)$
$P(X < 6) = 0.0194$
$P(X \leq 7) = 0.0453$
$P(X \geq 18) = 0.0551$
$P(X \geq 19) = 0.0287$ | M1; A1 A1 | (3) | M1 for writing or using $B(50, 0.25)$ also may be implied by both CR being correct. Condone use of P in critical region for the method mark. **A1** ($X \leq 6$) o.e. [0.6] DO NOT accept $P(X < 6)$; **A1** ($X \geq 19$) o.e. [19,50] DO NOT accept $P(X > 19)$
CR $X \leq 6$ and $X \geq 19$ | | |
**(c)** $0.0194 + 0.0287 = 0.0481$ | M1A1 | (2) | M1 Adding two probabilities for two tails. Both probabilities must be less than 0.5; A1 awrt 0.0481
**(d)** 8(It) is not in the Critical region or 8(It) is not significant or 0.0916 > 0.025; There is evidence that the probability of a faulty bolt is 0.25 or the company's claim is correct. | M1; A1ft | (2) | M1 one of the given statements followed through from their CR. **A1** contextual comment followed through from their CR. **NB** A correct contextual comment alone followed through from their CR, will get M1 A1
**(e)** $H_0: p = 0.25$ $H_1: p < 0.25$ | B1B1; M1A1 | (6) | B1 for $H_0$ must use $p$ or $\pi$ (pi); B1 for $H_1$ must use $p$ or $\pi$ (pi); M1 for finding or writing $P(X \leq 5)$ or attempting to find a critical region or a correct critical region; **A1** awrt 0.007/CR $X \leq 5$; 0.007 < 0.01, 5 is in the critical region, reject $H_0$, significant. There is evidence that the probability of faulty bolts has decreased. | M1; A1ft | 6) |
**Notes**
**(a)** B1 B1 one mark for each of any of the four statements. Give first B1 if only one correct statement given. No context needed.
**(b)** M1 for writing or using $B(50, 0.25)$ also may be implied by both CR being correct. Condone use of P in critical region for the method mark. **A1** ($X \leq 6$) o.e. [0.6] DO NOT accept $P(X < 6)$; **A1** ($X \geq 19$) o.e. [19,50] DO NOT accept $P(X > 19)$
**(c)** M1 Adding two probabilities for two tails. Both probabilities must be less than 0.5; A1 awrt 0.0481
**(d)** M1 one of the given statements followed through from their CR. **A1** contextual comment followed through from their CR. **NB** A correct contextual comment alone followed through from their CR, will get M1 A1
**(e)** B1 for $H_0$ must use $p$ or $\pi$ (pi); B1 for $H_1$ must use $p$ or $\pi$ (pi); M1 for finding or writing $P(X \leq 5)$ or attempting to find a critical region or a correct critical region; A1 awrt 0.007/CR $X \leq 5$; 0.007 < 0.01, 5 is in the critical region, reject $H_0$, significant. There is evidence that the probability of faulty bolts has decreased. **NB** A correct contextual statement alone followed through from their prob and $H_1$ get M1 A1
---A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.
\begin{enumerate}[label=(\alph*)]
\item Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample. [2]
\item Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is $\frac{1}{4}$. The probability of rejection in either tail should be as close as possible to 0.025 [3]
\item Find the actual significance level of this test. [2]
\end{enumerate}
In the sample of 50 the actual number of faulty bolts was 8.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Comment on the company's claim in the light of this value. Justify your answer. [2]
\end{enumerate}
The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Test at the 1\% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2010 Q6 [15]}}