| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 12 |
| 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 critical region calculation and a Poisson test. Part (a) involves routine binomial table lookup for two-tailed critical values, part (b) requires summing tail probabilities, and part (c) is a straightforward one-tailed Poisson test. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
| Answer | Marks | Guidance |
|---|---|---|
| \(X\) = no. of vases with defects; \(X \sim B(20, 0.15)\) | B1 | |
| \(P(X \leq 0) = 0.0388\) | M1 | Use of tables to find each tail |
| \(P(X \leq 6) = 0.9781 \therefore P(X \geq 7) = 0.0219\) | M1 | |
| \(\therefore\) critical region is \(X \leq 0\), or \(X \geq 7\) | A1, A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Significance level \(= P(X \leq 0) + P(X \geq 7) = 0.0388 + 0.0219 = 0.0607\) | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \lambda = 2.5\), \(H_1: \lambda > 2.5\) [or \(H_0: \lambda = 10\), \(H_1: \lambda > 10\)] | B1, B1 | |
| \(Y\) = no. sold in 4 weeks. Under \(H_0\): \(Y \sim Po(10)\) | M1 | |
| \(P(Y \geq 15) = 1 - P(Y \leq 14) = 1 - 0.9165 = 0.0835\) | M1, A1 | |
| More than 5% so not significant. Insufficient evidence of an increase in the rate of sales. | A1 | (6) |
## Question 4:
### Part (a):
| $X$ = no. of vases with defects; $X \sim B(20, 0.15)$ | B1 | |
|---|---|---|
| $P(X \leq 0) = 0.0388$ | M1 | Use of tables to find each tail |
| $P(X \leq 6) = 0.9781 \therefore P(X \geq 7) = 0.0219$ | M1 | |
| $\therefore$ critical region is $X \leq 0$, or $X \geq 7$ | A1, A1 | (5) |
### Part (b):
| Significance level $= P(X \leq 0) + P(X \geq 7) = 0.0388 + 0.0219 = 0.0607$ | B1 | (1) |
|---|---|---|
### Part (c):
| $H_0: \lambda = 2.5$, $H_1: \lambda > 2.5$ [or $H_0: \lambda = 10$, $H_1: \lambda > 10$] | B1, B1 | |
|---|---|---|
| $Y$ = no. sold in 4 weeks. Under $H_0$: $Y \sim Po(10)$ | M1 | |
| $P(Y \geq 15) = 1 - P(Y \leq 14) = 1 - 0.9165 = 0.0835$ | M1, A1 | |
| More than 5% so not significant. Insufficient evidence of an increase in the rate of sales. | A1 | (6) |
---4. From past records a manufacturer of glass vases knows that $15 \%$ of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.
\begin{enumerate}[label=(\alph*)]
\item Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to $2.5 \%$.
\item State the actual significance level of the test described in part (a).
A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
\item Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence that the rate of sales per week had increased in December.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [12]}}