At a certain large school it was found that the proportion of students not wearing correct uniform was 0.15. The school sent a letter to parents asking them to ensure that their children wear the correct uniform. The school now wishes to test whether the proportion not wearing correct uniform has been reduced.

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

1. [B1] Not representative (of all students in the school). OE idea of 'not being representative' e.g. different grades in the school have different characteristics/proportions.... Don't accept 'not random' or 'biased' without further explanation.
2. [B1] Null Hypothesis `H₀: p = 0.15` and Alternative Hypothesis `H₁: p < 0.15`. Allow "p" instead of `P(not correct uniform)`.
3. [M1] For attempting any two probabilities using `B(50, 0.15)`.
4. [M1] For calculating `P(X ≤ 3)` using `B(50, 0.15)`. Attempt the tail probability `P(0,1,2,3)` with `B(50,0.15)` must be added.
5. [M1] For calculating `P(X ≤ 4)` using `B(50, 0.15)`. OE. Their `P(X ≤ 3) + P(X= 4)` or `P(0,1,2,3,4)` with `B(50,0.15)` must be added.
6. [A1] For stating both `P(X ≤ 3) = 0.0460` (or 0.0461) and `P(X ≤ 4) = 0.112` (or correct values showing `<0.05` and `>0.05` respectively). OR if `P(X ≤ 4)` not seen; `P(4)=0.06606` and `0.06606 > 0.05` and `P(X ≤ 3)=0.0460` scores M1 A1.
7. [A1] For stating `P(Type I error) = 0.0460` (or 0.0461) (3 sf). Dependent on second M1. SC If M1M1M1A0 scored allow A1FT for incorrect `P(X ≤ 3)` as long as `<0.05`.
8. [M1] For conclusion based on comparison: Observed value `4` is outside critical region `(X ≤ 3)` OE or `P(X ≤ 4) = 0.112` which is `> 0.05`. FT working from (c).
9. [A1] For stating conclusion in context: No evidence that proportion not wearing the correct uniform has decreased (Accept H₀). In context not definite, e.g. not 'Proportion has not decreased'. No contradiction.
10. [*B1] Type of error: Type II error may have been made. FT If Reject H₀ in (d).
11. [DB1] Reason: H₀ was not rejected, but it might actually be false (the proportion actually decreased, but the sample did not provide enough evidence). FT Type I if H₀ was rejected.