| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2008 |
| Session | June |
| 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 context. Students must identify H₀: p=0.35 vs H₁: p>0.35, calculate P(X≥8) where X~B(15,0.35), and compare to 5%. The setup is standard and the calculations are routine for S1, though the cumulative probability calculation requires care. Slightly easier than average due to clear signposting and being a textbook application. |
| Spec | 2.05a 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 |
|---|---|---|
| \(H_1: p > 0.35\) | B1, B1, B1 | B1 for definition of \(p\); B1 for \(H_0\); B1 for \(H_1\) |
| \(H_1\) has this form since the student believes that the probability will be increased/improved/got better/gone up. | E1dep | E1dep on \(p > 0.35\) in \(H_0\) in words not just because \(p > 0.35\) |
| Answer | Marks | Guidance |
|---|---|---|
| Either: \(P(X \geq 8) = 1 - 0.8868 = 0.1132 > 5\%\) | M1, M1dep, A1dep | Either: M1 for probability (0.1132); M1dep for comparison; A1dep |
| Answer | Marks | Guidance |
|---|---|---|
| So not enough evidence to reject \(H_0\) (Accept \(H_0\)) | E1dep | E1dep on all previous marks for conclusion in context |
| Answer | Marks | Guidance |
|---|---|---|
| Conclude that there is not enough evidence to indicate that the probability of remembering all of the items is improved/improved/got better/gone up. (when listening to music.) | E1dep | E1dep on all previous marks for conclusion in context |
| Answer | Marks | Guidance |
|---|---|---|
| Conclude that there is not enough evidence to indicate that the probability of remembering all of the items is improved (when listening to music) | E1dep | E1dep on all previous marks for conclusion in context |
## (i)
Let $p =$ probability of remembering or naming all items (for population) (whilst listening to music.)
$H_0: p = 0.35$
$H_1: p > 0.35$ | B1, B1, B1 | B1 for definition of $p$; B1 for $H_0$; B1 for $H_1$
$H_1$ has this form since the student believes that the probability will be increased/improved/got better/gone up. | E1dep | E1dep on $p > 0.35$ in $H_0$ in words not just because $p > 0.35$ | 4 marks total
## (ii)
Let $X \sim B(15, 0.35)$
**Either:** $P(X \geq 8) = 1 - 0.8868 = 0.1132 > 5\%$ | M1, M1dep, A1dep | Either: M1 for probability (0.1132); M1dep for comparison; A1dep
Or $0.8868 < 95\%$
So not enough evidence to reject $H_0$ (Accept $H_0$) | E1dep | E1dep on all previous marks for conclusion in context
---
**Or:**
Critical region for the test is $\{9, 10, 11, 12, 13, 14, 15\}$
8 does not lie in the critical region.
So not enough evidence to reject $H_0$
Conclude that there is not enough evidence to indicate that the probability of remembering all of the items is improved/improved/got better/gone up. (when listening to music.) | E1dep | E1dep on all previous marks for conclusion in context
---
**Or:**
The smallest critical region that 8 could fall into is $\{8, 9, 10, 11, 12, 13, 14, 15\}$. The size of this region is 0.1132
$0.1132 \geq 5\%$
So not enough evidence to reject $H_0$
Conclude that there is not enough evidence to indicate that the probability of remembering all of the items is improved (when listening to music) | E1dep | E1dep on all previous marks for conclusion in context | 4 marks for this part; 8 marks total
---5 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 2008 Q5 [8]}}