IBDP Maths SL 4.1 Concepts of population, sample AA HL Paper 2- Exam Style Questions- New Syllabus
Question
(ii) Write down the mode of \( X \).
Most-appropriate topic codes (IB Mathematics: Analysis and Approaches HL 2025):
• SL 4.5: Probability of events, expected value — part (b)
• SL 4.7: Discrete random variables, expected value — part (b)
• SL 4.1: Sampling techniques — part (c)
▶️ Answer/Explanation
(a)(i)
The sum of all probabilities in a distribution must equal 1.
\( 1.5a + 2a + 0.281 + a + 0.026 = 1 \)
\( 4.5a + 0.307 = 1 \)
\( 4.5a = 0.693 \)
\( a = \frac{0.693}{4.5} = 0.154 \)
\( \boxed{a = 0.154} \)
(a)(ii)
Mode is the value of \( x \) with the highest probability.
Probabilities: \( P(X=1) = 1.5 \times 0.154 = 0.231 \) \( P(X=2) = 2 \times 0.154 = 0.308 \) \( P(X=3) = 0.281 \) \( P(X=4) = 0.154 \) \( P(X=5) = 0.026 \)
The highest probability is \( 0.308 \) when \( x = 2 \).
\( \boxed{\text{Mode} = 2} \)
(b)
Mean \( = E(X) = \sum x \cdot P(X=x) \):
\( E(X) = 1 \times 0.231 + 2 \times 0.308 + 3 \times 0.281 + 4 \times 0.154 + 5 \times 0.026 \)
\( = 0.231 + 0.616 + 0.843 + 0.616 + 0.13 \)
\( = 2.436 \)
To three significant figures:
\( \boxed{\text{Mean} = 2.44} \)
(c)
The manager surveys the first 50 customers to arrive—this is easily accessible and based on convenience, not random selection.
\( \boxed{\text{Convenience sampling}} \)