| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| 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 standard structure: state hypotheses, define parameter, and perform the test using normal approximation or tables. All steps are routine AS-level procedures with no conceptual challenges beyond applying the standard method to given data. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05f Pearson correlation coefficient |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.37\) and \(H_1: p < 0.37\) | B1 | AO 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(p\) is the probability that an adult selected at random in the United Kingdom never exercises (or plays sport) | B1 | AO 2.5 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 35) = 0.058\ldots\) | B1 | AO 1.1 |
| Their \(0.058\ldots\) compared with \(0.05\) | M1 | AO 1.1 |
| \(0.058 > 0.05\) or \(35.033\ldots > 35\) or \(0.059867 > 0.05\) and 'so do not reject \(H_0\)' | A1 | AO 2.2b |
| There is no evidence (or insufficient evidence) to suggest/support at the 5% level that the percentage of adults (selected at random in the United Kingdom) who never exercises (or plays sport) is less than 37% | A1\* | AO 2.4 |
## Question 13:
### Part (a):
$H_0: p = 0.37$ and $H_1: p < 0.37$ | **B1** | AO 1.1 | Allow equivalent in words. Do not allow percentages.
### Part (b):
$p$ is the probability that an adult selected at random in the United Kingdom never exercises (or plays sport) | **B1** | AO 2.5 | Accept 'proportion' but **not** number/amount etc. B1B1 in (a)(b) if another symbol used **if correctly defined**. Underlined words needed.
### Part (c):
$P(X \leq 35) = 0.058\ldots$ | **B1** | AO 1.1 | May see $P(X < 35) = 0.0386$ or $P(X \leq 36) = 0.0847$ using $X \sim B(118, 0.37)$ — SC 1 mark. Use of $P(X = 35) = 0.01961$ is **M0**. Normal approx: $Y \sim N(43.66, 27.5058)$ gives CV $35.033\ldots$; $P(Y \leq 35.5) = 0.059867\ldots$ (must use continuity correction); $P(Y \leq 35) = 0.0493465\ldots$ is M0
Their $0.058\ldots$ compared with $0.05$ | **M1** | AO 1.1 |
$0.058 > 0.05$ or $35.033\ldots > 35$ or $0.059867 > 0.05$ and 'so do not reject $H_0$' | **A1** | AO 2.2b | 'accept $H_0$' is ok
There is **no evidence (or insufficient evidence)** to **suggest/support** at the 5% level that the **percentage** of **adults** (selected at random in the United Kingdom) who **never exercises** (or plays sport) **is less than 37%** | **A1\*** | AO 2.4 | Fully correct contextual conclusion. No assertive statements. Accept percentage/proportion or probability with 0.37. Dependent on award of all other marks in (c).
---13 In a report published in October 2021 it is stated that $37 \%$ of adults in the United Kingdom never exercise or play sport. A researcher believes that the true percentage is less than this. They decide to carry out a hypothesis test at the $5 \%$ level to investigate the claim.
\begin{enumerate}[label=(\alph*)]
\item State the null and alternative hypotheses for their test.
\item Define the parameter for their test.
In a random sample of 118 adults, they find that 35 of them never exercise or play sport.
\item Carry out the test.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2023 Q13 [6]}}