| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Standard +0.3 This is a straightforward one-tailed binomial hypothesis test with clear hypotheses (p = 0.097 vs p > 0.097), large sample size allowing normal approximation, and standard procedure. Part (a) requires basic understanding of binomial assumptions. The calculation is routine with no conceptual challenges, making it slightly easier than average. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. The patients (or cases) are independent oe e.g. Patients form random sample of the population | B1 [1] | oe, in context; Allow any indication of context eg "patient" or "side effects"; NOT "Prob of patients getting side effects is indep or random"; Ignore all else |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: p = 0.097\) where \(p\) is the proportion of patients experiencing side-effects (within a year) or 9.7% of patients experience side-effects | B1 | Allow "possibility" or "probability"; Not \(p\) = percentage having disease |
| \(H_1: p > 0.097\) or \(> 9.7\%\) exp side-effects | B1 | One error e.g. undefined \(p\), \(p = 9.7\%\) scores B1B0 |
| \(B(450, 0.097)\) & \(X = 50\) (allow 51) | M1 | Stated or implied e.g. by 0.138 or 0.107 or 0.108 or 0.862 or 0.893 or 0.0308 or 0.0366; NB 0.138 seen (or 0.107 or 0.108 or 0.862 or 0.893) implies M1 even if part of incorrect statement; e.g. \(P(X \leq 51) = 0.107\) or \(P(X > 51) = 0.138\) |
| \(P(X \geq 51) = 1 - 0.862 = 0.138\) (3 sf) | A1 | cao BC |
| Compare 0.1 | A1f | Dep 0.138 or 0.107 or 0.108 only |
| Insufficient evidence to reject \(H_0\); Allow "Not reject \(H_0\)"; Allow Accept \(H_0\) | M1 | Must see this statement; Dep 0.138 or 0.107 or 0.108 or \(P(X \geq 51\) or 50) stated |
| No evidence (at 10% level) that proportion experiencing side-effects in one year under new treatment is greater than under standard treatment | A1f | Any equivalent statement, in context, not definite; allow "likelihood", "percentage"; Ignore all else; ft only their \(P(X \geq 51)\); NB possible opposite conclusion on ft |
| Answer | Marks |
|---|---|
| \(H_0: p = 0.097\) (defined \(p\)) | B1 |
| \(H_1: p \neq 0.097\) | B0 |
| Compare 0.05 | A1; No more marks |
## Question 11:
### Part (a):
e.g. The probability of side effects is the same for each patient (or is constant)
e.g. The patients (or cases) are independent oe e.g. Patients form random sample of the population | **B1** [1] | oe, in context; Allow any indication of context eg "patient" or "side effects"; **NOT** "Prob of patients getting side effects is indep or random"; Ignore all else
# Question 11(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.097$ where $p$ is the proportion of patients experiencing side-effects (within a year) or 9.7% of patients experience side-effects | **B1** | Allow "possibility" or "probability"; Not $p$ = percentage having disease |
| $H_1: p > 0.097$ or $> 9.7\%$ exp side-effects | **B1** | One error e.g. undefined $p$, $p = 9.7\%$ scores B1B0 |
| $B(450, 0.097)$ & $X = 50$ (allow 51) | **M1** | Stated or implied e.g. by 0.138 or 0.107 or 0.108 or 0.862 or 0.893 or 0.0308 or 0.0366; NB 0.138 seen (or 0.107 or 0.108 or 0.862 or 0.893) implies M1 even if part of incorrect statement; e.g. $P(X \leq 51) = 0.107$ or $P(X > 51) = 0.138$ |
| $P(X \geq 51) = 1 - 0.862 = 0.138$ (3 sf) | **A1** | cao **BC** |
| Compare 0.1 | **A1f** | Dep 0.138 or 0.107 or 0.108 only |
| Insufficient evidence to reject $H_0$; Allow "Not reject $H_0$"; Allow Accept $H_0$ | **M1** | Must see this statement; Dep 0.138 or 0.107 or 0.108 or $P(X \geq 51$ or 50) stated |
| No evidence (at 10% level) that proportion experiencing side-effects in one year under new treatment is greater than under standard treatment | **A1f** | Any equivalent statement, in context, not definite; allow "likelihood", "percentage"; Ignore all else; ft only their $P(X \geq 51)$; NB possible opposite conclusion on ft |
**If 2-tail test:**
| $H_0: p = 0.097$ (defined $p$) | **B1** |
| $H_1: p \neq 0.097$ | **B0** |
| Compare 0.05 | **A1**; No more marks |
---11 It is known that, under the standard treatment for a certain disease, $9.7 \%$ of patients with the disease experience side effects within one year.
In a trial of a new treatment, a random sample of 450 patients with this disease was selected and the number $X$ who experienced side effects within one year was noted.
\begin{enumerate}[label=(\alph*)]
\item State one assumption needed in order to use a binomial model for $X$.
It was found that 51 of the 450 patients experienced side effects within one year.
\item Test, at the $10 \%$ significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q11 [8]}}