IB Mathematics SL 4.1 Concepts of population, sample AI HL Paper 2- Exam Style Questions- New Syllabus
▶️ Answer/Explanation
(a)(i)
Order the scores (ascending): 302, 305, 331, 334, 364, 370, 472, 498
With \( n = 8 \), use the median-of-halves rule:
Upper half = (364, 370, 472, 498) \( \Rightarrow Q_3 = \frac{370 + 472}{2} = 421 \) (A1)
Result:
\( Q_3 = 421 \)
(a)(ii)
Lower half = (302, 305, 331, 334) \( \Rightarrow Q_1 = \frac{305 + 331}{2} = 318 \)
\( IQR = Q_3 – Q_1 = 421 – 318 = 103 \) (A1)
Result:
\( IQR = 103 \)
(b)
Upper fence: \( Q_3 + 1.5 \times IQR = 421 + 1.5 \times 103 = 575.5 \)
Lower fence: \( Q_1 – 1.5 \times IQR = 318 – 1.5 \times 103 = 163.5 \)
Netherlands’ score \( 498 \) lies between 163.5 and 575.5 \( \Rightarrow \) not an outlier (A1)(A1)
Result:
Netherlands is not an outlier
(c)
Pearson’s \( r = 0.249 \) is close to 0 \( \Rightarrow \) weak linear association, so a regression line is not appropriate for prediction (A1)
Result:
Not appropriate (r too close to zero)
(d)(i)
Population ranks (to the nearest million): Russia 1, Italy 2, Netherlands 3, Sweden 4.5, Azerbaijan 4.5, Switzerland 6, Norway 7, North Macedonia 8
For Switzerland (9 million), \( a = 6 \) (A1)
Result:
\( a = 6 \)
(d)(ii)
For Sweden (10 million), \( b = 4.5 \) (tied with Azerbaijan) (A1)
Result:
\( b = 4.5 \)
(d)(iii)
For Azerbaijan (10 million), \( c = 4.5 \) (tied with Sweden) (A1)
Result:
\( c = 4.5 \)
(e)(i)
Using average ranks for ties and computing Spearman’s \( r_s \) (Pearson on ranks) gives
\( r_s \approx 0.683 \) (to 3 d.p.) (A1)
Result:
\( r_s = 0.683 \)
(e)(ii)
There is a positive association between population and Eurovision score ranks (A1)
Result:
Positive association between population and score ranks
(f)
Reading Netherlands as 478 instead of 498 does not change its rank (still the highest)
Since Spearman uses ranks, all rank differences — and thus \( r_s \) — are unchanged (A1)
Result:
Netherlands remains top rank, so \( r_s \) unchanged