| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | January |
| Marks | 13 |
| 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 routine application of binomial distribution tables to find critical regions and perform a significance test. Part (a) involves straightforward table lookup for critical values, part (b) is trivial recall, and part (c) is a standard one-tailed test with clear hypotheses. While multi-part with 13 marks total, it requires no novel insight—just methodical application of learned procedures, making it slightly easier than the average A-level question. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(X\) represent the number of plant pots with defects, \(X \sim B(25, 0.20)\) | Implied B1 | |
| \(P(X \leq 1) = 0.0274\), \(P(X \geq 10) = 0.0173\) | Clear attempt at both tails required, 4dp M1A1A1 | |
| Critical region is \(X \leq 1, X \geq 10\) | A1 |
| Answer | Marks |
|---|---|
| Significance level \(= 0.0274 + 0.0173 = 0.0447\) | Accept % 4dp B1 cao |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \lambda = 10\), \(H_1: \lambda > 10\) (or \(H_0: \lambda = 60\), \(H_1: \lambda > 60\)) | B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(Y \geq 74) = P(W > 73.5)\) where \(W \sim N(60, 60)\) | \(\pm 0.5\) for cc, 73.5 M1A1 | |
| \(\approx P\left(Z \geq \frac{73.5 - 60}{\sqrt{60}}\right) = P(Z > 1.74) = 0.0407 - 0.0409 < 0.05\) | Standardise using \(60\sqrt{60}\) M1, A1 | |
| Evidence that rate of sales per week has increased. | A1J |
## Part (a)
Let $X$ represent the number of plant pots with defects, $X \sim B(25, 0.20)$ | Implied B1 |
$P(X \leq 1) = 0.0274$, $P(X \geq 10) = 0.0173$ | Clear attempt at both tails required, 4dp M1A1A1 |
Critical region is $X \leq 1, X \geq 10$ | | A1 |
(5 marks)
## Part (b)
Significance level $= 0.0274 + 0.0173 = 0.0447$ | Accept % 4dp B1 cao |
(1 mark)
## Part (c)
$H_0: \lambda = 10$, $H_1: \lambda > 10$ (or $H_0: \lambda = 60$, $H_1: \lambda > 60$) | | B1B1 |
Let $Y$ represent the number sold in 6 weeks, under $H_0$, $Y \sim \text{Po}(60)$
$P(Y \geq 74) = P(W > 73.5)$ where $W \sim N(60, 60)$ | $\pm 0.5$ for cc, 73.5 M1A1 |
$\approx P\left(Z \geq \frac{73.5 - 60}{\sqrt{60}}\right) = P(Z > 1.74) = 0.0407 - 0.0409 < 0.05$ | Standardise using $60\sqrt{60}$ M1, A1 |
Evidence that rate of sales per week has increased. | | A1J |
(7 marks)
**Total 13 Marks**
---From past records a manufacturer of ceramic plant pots knows that 20\% of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.
\begin{enumerate}[label=(\alph*)]
\item Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20. The probability of rejection in either tail should be as close as possible to 2.5\%. [5]
\item Write down the significance level of the above test. [1]
\end{enumerate}
A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Using a 5\% level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2004 Q6 [13]}}