| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated parameters (n=15, p=0.35, x=8). Students must state hypotheses correctly, calculate P(X≥8) using binomial tables or formula, and compare to 5% significance level. While it requires understanding of hypothesis testing framework, the calculations are routine and the context is clear, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Let \(p\) = probability of remembering/naming all items (for population) whilst listening to music | B1 | for definition of \(p\) |
| \(H_0: p = 0.35\) | B1 | for \(H_0\) |
| \(H_1: p > 0.35\) | B1 | for \(H_1\) |
| \(H_1\) has this form since the student believes the probability will be increased/improved | E1dep | on \(p>0.35\) in \(H_0\); in words, not just because \(p > 0.35\) — Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Let \(X \sim B(15, 0.35)\) | ||
| Either: \(P(X \geq 8) = 1 - 0.8868 = 0.1132 > 5\%\) or \(0.8868 < 95\%\) | M1 | for probability (0.1132) |
| So not enough evidence to reject \(H_0\) | M1dep | for comparison |
| A1dep | ||
| Conclude: not enough evidence that probability of remembering all items is improved when listening to music | E1dep | on all previous marks, conclusion in context |
| Or: Critical region for the test is \(\{9,10,11,12,13,14,15\}\); 8 does not lie in CR | M1 | for correct CR (no omissions or additions) |
| So not enough evidence to reject \(H_0\) | M1dep | for 8 does not lie in CR |
| A1dep | ||
| Conclude in context | E1dep | on all previous marks — Total: 4 |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $p$ = probability of remembering/naming all items (for population) whilst listening to music | B1 | for definition of $p$ |
| $H_0: p = 0.35$ | B1 | for $H_0$ |
| $H_1: p > 0.35$ | B1 | for $H_1$ |
| $H_1$ has this form since the student believes the probability will be increased/improved | E1dep | on $p>0.35$ in $H_0$; in words, not just because $p > 0.35$ — **Total: 4** |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $X \sim B(15, 0.35)$ | | |
| **Either:** $P(X \geq 8) = 1 - 0.8868 = 0.1132 > 5\%$ or $0.8868 < 95\%$ | M1 | for probability (0.1132) |
| So not enough evidence to reject $H_0$ | M1dep | for comparison |
| | A1dep | |
| Conclude: not enough evidence that probability of remembering all items is improved when listening to music | E1dep | on all previous marks, conclusion in context |
| **Or:** Critical region for the test is $\{9,10,11,12,13,14,15\}$; 8 does not lie in CR | M1 | for correct CR (no omissions or additions) |
| So not enough evidence to reject $H_0$ | M1dep | for 8 does not lie in CR |
| | A1dep | |
| Conclude in context | E1dep | on all previous marks — **Total: 4** |
---3 A psychology student is investigating memory. In an experiment, volunteers are given 30 seconds to try to memorise a number of items. The items are then removed and the volunteers have to try to name all of them. It has been found that the probability that a volunteer names all of the items is 0.35 . The student believes that this probability may be increased if the volunteers listen to the same piece of music while memorising the items and while trying to name them.
The student selects 15 volunteers at random to do the experiment while listening to music. Of these volunteers, 8 name all of the items.\\
(i) Write down suitable hypotheses for a test to determine whether there is any evidence to support the student's belief, giving a reason for your choice of alternative hypothesis.\\
(ii) Carry out the test at the $5 \%$ significance level.
\hfill \mbox{\textit{OCR MEI S1 Q3 [8]}}