| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | State hypotheses with additional parts |
| Difficulty | Moderate -0.3 This is a standard hypothesis testing question covering routine procedures (stating hypotheses, finding critical regions, conducting a one-tailed binomial test). While it has multiple parts, each part follows textbook methodology with no novel problem-solving required. The calculations are straightforward using normal approximation to binomial, making it slightly easier than average for AS-level statistics. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail2.05f Pearson correlation coefficient |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.81\) and \(H_1: p < 0.81\) | B1 | \(H_0: probability = 0.81\); \(H_1: probability < 0.81\) |
| \(p\) is the probability that a young adult (selected at random in England) has never donated blood | B1 | Or proportion; NOT number or how many young adults has never donated blood |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.05\) | B1 | Accept 5%, 1/20 oe |
| Answer | Marks | Guidance |
|---|---|---|
| \(324\) | B1 | \(400\times0.81 = 324\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.026 - 0.026125\) BC | B1 | Percentages acceptable |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 310) = 0.0448 - 0.045\ [< 0.05]\) | \*M1 | Additional calculations not penalised; SC If no marks scored and state \(x \leq 310\) or \(x<311\): B1 |
| \(P(X \leq 311) = 0.0576 - 0.058\ [> 0.05]\) | \*M1 | |
| hence CR is \([0 \leq]\ x \leq 310\) | DA1 | Accept \(x<311\) |
| Answer | Marks | Guidance |
|---|---|---|
| 314 is not in critical region | M1 | FT Comparison of 314 with *their* CR; or \(P(X \leq 314) = 0.114 > 0.05\); SC If prob incorrect but correct comparison and consistent conclusion: M1A0A0 |
| accept \(H_0\) | A1 | FT consistent with M mark; allow "not significant" / "reject \(H_1\)"; If incorrect CR then could get M1A1A0 |
| there is insufficient evidence at the 5% level to suggest that the percentage of young adults (in England who have never given blood) is less than 81% oe | A1 | Not a definite statement, not prove; Oe. Not enough evidence to suggest that the campaign has made more young people donate blood. |
## Question 8(a):
$H_0: p = 0.81$ and $H_1: p < 0.81$ | **B1** | $H_0: probability = 0.81$; $H_1: probability < 0.81$
$p$ is the probability that a young adult (selected at random in England) has never donated blood | **B1** | Or proportion; NOT number or how many young adults has never donated blood
## Question 8(b):
$0.05$ | **B1** | Accept 5%, 1/20 oe
## Question 8(c):
$324$ | **B1** | $400\times0.81 = 324$
## Question 8(d):
$0.026 - 0.026125$ **BC** | **B1** | Percentages acceptable
## Question 8(e):
$P(X \leq 310) = 0.0448 - 0.045\ [< 0.05]$ | **\*M1** | Additional calculations not penalised; SC If no marks scored and state $x \leq 310$ or $x<311$: B1
$P(X \leq 311) = 0.0576 - 0.058\ [> 0.05]$ | **\*M1** |
hence CR is $[0 \leq]\ x \leq 310$ | **DA1** | Accept $x<311$
## Question 8(f):
314 is not in critical region | **M1** | FT Comparison of 314 with *their* CR; or $P(X \leq 314) = 0.114 > 0.05$; SC If prob incorrect but correct comparison and consistent conclusion: M1A0A0
accept $H_0$ | **A1** | FT consistent with M mark; allow "not significant" / "reject $H_1$"; If incorrect CR then could get M1A1A0
there is insufficient evidence at the 5% level to **suggest** that the percentage of young adults (in England who have never given blood) is less than 81% oe | **A1** | Not a **definite** statement, not prove; Oe. Not enough evidence to suggest that the campaign has made more young people donate blood.
---8 In 2018 research showed that 81\% of young adults in England had never donated blood.\\
Following an advertising campaign in 2021, it is believed that the percentage of young adults in England who had never donated blood in 2021 is less than $81 \%$.
Ling decides to carry out a hypothesis test at the 5\% level.\\
Ling collects data from a random sample of 400 young adults in England.
\begin{enumerate}[label=(\alph*)]
\item State the null and alternative hypotheses for the test, defining the parameter used.
\item Write down the probability that the null hypothesis is rejected when it should in fact be accepted.
\item Assuming the null hypothesis is correct, calculate the expected number of young adults in the sample who had never donated blood.
\item Calculate the probability that there were no more than 308 young adults who had never donated blood in the sample.
\item Determine the critical region for the test.
In fact, the sample contained 314 young adults who had never donated blood.
\item Carry out the test, giving the conclusion in the context of the question.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2022 Q8 [11]}}