In a large college, 28% of the students do not play any musical instrument, 52% play exactly one musical instrument and the remainder play two or more musical instruments. A random sample of 12 students from the college is chosen.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. M1 One term ¹2Cx pˣ (1-p)¹²⁻ˣ, for 0<x<12, 0<p<1.
2. A1 Correct expression, accept unsimplified, no terms omitted, leading to final answer.
3. B1 Final answer 0.3036 <p ≤ 0.304. [0.304]
4. M1 One term ¹2Cx pˣ (1-p)¹²⁻ˣ, for 0 < x < 12, 0<p<1.
5. A1 Correct expression, accept unsimplified, no terms omitted, leading to final answer.
6. B1 Final answer 0.3036 <p ≤ 0.304.
7. B1 46.8 and 22.464 or 22.46 seen, allow unsimplified, (4.739 < σ < 4.740 imply correct variance). [Mean = 0.52×90 = 46.8, var = 0.52×0.48×90 = 22.464]
8. M1 Substituting their mean and their variance into standardisation formula (any number for 39.5), not σ², √ σ.
9. M1 Using continuity correction 39.5 or 40.5 in their standardisation formula.
10. M1 Appropriate area Φ, from final process, must be probability.
11. A1 0.06175 < p < 0.0618. [0.0618]