| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (lower tail, H₁: p < p₀) |
| Difficulty | Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated context and parameters. Students must state H₀: p=0.2 vs H₁: p<0.2, calculate P(X≤3) where X~B(40,0.2), and compare to 5% significance level. While it requires proper hypothesis test structure and binomial probability calculation, it's a standard S2 textbook exercise with no conceptual surprises—slightly easier than average due to small critical value making calculations manageable. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: p = 0.2 \quad H_1: p < 0.2\) | B1 | Must use \(p\) or \(\pi\) for both \(H_0\) and \(H_1\) |
| \([X \sim B(40, 0.2)]\) \(\quad P(X \leq 3) = 0.0285\) or CR of \(X \leq 3\) | M1A1 | M1 for writing or using \(B(40, 0.2)\), may be implied by correct answer; A1 awrt \(0.0285\) or CR of \(X \leq 3\) as final answer |
| \([0.0285 < 0.05]\) significant, reject \(H_0\) | M1dep | Dependent on previous M mark; correct contextual comparison of probability with \(0.05\) (or comparing \(3\) with critical region); no conflicting statements |
| There is evidence to support the supplier's claim, or: the probability of a ball failing the bounce test is less than 0.2 | A1cso | This is cso — only awarded for fully correct solution; correct contextualised conclusion must include the underlined bold words |
| Total | (5) |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.2 \quad H_1: p < 0.2$ | B1 | Must use $p$ or $\pi$ for both $H_0$ and $H_1$ |
| $[X \sim B(40, 0.2)]$ $\quad P(X \leq 3) = 0.0285$ or CR of $X \leq 3$ | M1A1 | M1 for writing or using $B(40, 0.2)$, may be implied by correct answer; A1 awrt $0.0285$ or CR of $X \leq 3$ as final answer |
| $[0.0285 < 0.05]$ significant, reject $H_0$ | M1dep | Dependent on previous M mark; correct contextual comparison of probability with $0.05$ (or comparing $3$ with critical region); no conflicting statements |
| There is evidence to support the supplier's **claim**, or: the probability of a **ball** failing the **bounce test** is **less** than **0.2** | A1cso | This is cso — only awarded for fully correct solution; correct contextualised conclusion must include the underlined bold words |
| **Total** | **(5)** | |\begin{enumerate}
\item Before Roger will use a tennis ball he checks it using a "bounce" test. The probability that a ball from Roger's usual supplier fails the bounce test is 0.2 . A new supplier claims that the probability of one of their balls failing the bounce test is less than 0.2 . Roger checks a random sample of 40 balls from the new supplier and finds that 3 balls fail the bounce test.
\end{enumerate}
Stating your hypotheses clearly, use a $5 \%$ level of significance to test the new supplier's claim.\\
\hfill \mbox{\textit{Edexcel S2 2014 Q1 [5]}}