June 2024 Paper 3 Q4
4. The proportion of left-handed adults in a country is 10%
Freya believes that the proportion of left-handed adults under the age of 25 in this country is different from 10%
She takes a random sample of 40 adults under the age of 25 from this country to investigate her belief.
You should
- state your hypotheses clearly
- use a 5% level of significance
- state the probability of rejection in each tail
In Freya’s sample 7 adults were left-handed.
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H_0}: p = 0.1 \qquad \mathrm{H_1}: p \neq 0.1\) [Allow 10% for 0.1] | B1 | 2.5 |
| [\(X \sim \mathrm{B}(40, 0.1)\)] \(\Rightarrow \mathrm{P}(X = 0) = 0.0148\) [Allow any letter for \(X\)] | M1 | 3.3 |
| \(\mathrm{P}(X \geqslant 9) = 1 - \mathrm{P}(X \leqslant 8) = 1 - 0.9845 = 0.0155\) | A1 | 1.1b |
| Critical region is \(\underline{\{X = 0\} \cup \{X \geqslant 9\}}\) (o.e.) | A1 | 1.1b |
| (4) |
Notes
Mark (a) and (b) together for sight of the probabilities.
B1 for both hypotheses in terms of \(p\) or \(\pi\). Must be attached to \(\mathrm{H_0}\) and \(\mathrm{H_1}\)
M1 for use of the correct model. Implied by sight of at least one probability truncated or rounded to at least 2sf from: 0.0155, 0.0148, 0.0805, 0.9845, 0.9581, 0.9949
Implied by sight of fully correct CR (with no probs) so e.g. \(X = 0, X \gt 8\) scores M1A0A0
1st A1 for at least one correct probability (to at least 3sf) with its probability statement
i.e. for \(\mathrm{P}(X = 0) =\) awrt 0.0148 or \(\mathrm{P}(X \geqslant 9) =\) awrt 0.0155
2nd A1 (dep on M1 and 1st A1 but not on B1)
for both correct probs (to at least 3 sf) and the correct critical region
Do not need set notation. Allow \(X \lt 1\) and \(X \gt 8\) or words e.g. “0 or greater than 8” etc
Allow “,” or “and” or “or” or “\(\cap\)” between \(X \leqslant 0\) and \(X \geqslant 9\)
\(\mathrm{P}(X = 0)\) and \(\mathrm{P}(X \geqslant 9)\) is 2nd A0
| Scheme | Marks | AO |
|---|---|---|
| [“0.0148” + “0.0155”] = 0.0303 | B1ft | 1.1b |
| (1) |
Notes
B1ft for awrt 3.03% or correct sum of their two probabilities (provided each is less than 0.5)
Their probabilities must be to at least 2sf and relate to their CR
| Scheme | Marks | AO |
|---|---|---|
| [Provided 7 is not in their CR] insufficient evidence to support Freya’s belief | B1 | 2.2b |
| (1) | ||
| (6 marks) |
Notes
To score in (c) they must have a CR of the form (\(X = 0\) or \(X \lt a\)) and \(X \gt b\), where \(b\) is \(\geqslant 7\)
May be implied by \(\mathrm{P}(X \lt a)\) and \(\mathrm{P}(X \gt b)\) i.e. 2nd A0 in (a) but correct form. Need \(b \gt a\)
B1 for a suitable comment in context that suggests no support for Freya’s belief/claim
or e.g. insufficient evidence of change in proportion/percentage of left-handed adults
or e.g. proportion/percentage of left-handed adults is not different from 10% (or …is 10%)
or e.g. 10% of adults in the country are left-handed
Do not allow contradictory comments e.g. “in CR so no support for Freya’s belief” is B0
NB A correct contextual answer in (c) using an acceptance region please send to review.