IB Mathematics SL 4.3 Measures of central tendency AA SL Paper 2- Exam Style Questions- New Syllabus
Question
| \( x \) | 1 | 2 | 3 | 4 | 5 | \( \geq 6 \) |
|---|---|---|---|---|---|---|
| \( P(X = x) \) | \( 1.5a \) | \( 2a \) | 0.281 | \( a \) | 0.026 | 0 |
(ii) State the mode of \( X \).
(b) Calculate the expected value (mean) of \( X \).
(c) To collect data, the manager interviews the first 50 customers who enter the store on a specific day.
Relevant Syllabus Topics:
• SL 4.3: Measures of central tendency — parts (a)(ii), (b)
• SL 4.1: Sampling techniques — part (c)
▶️ Answer/Explanation
(a)(i)
The sum of all probabilities in the distribution must equal 1:
\( 1.5a + 2a + 0.281 + a + 0.026 = 1 \)
\( 4.5a + 0.307 = 1 \)
\( 4.5a = 0.693 \)
\( a = 0.154 \)
(a)(ii)
The mode is the outcome with the highest probability. [cite: 1321] Comparing values: \( P(X=1) = 0.231 \), \( P(X=2) = 0.308 \), \( P(X=3) = 0.281 \), \( P(X=4) = 0.154 \), \( P(X=5) = 0.026 \).
The maximum probability is \( P(X=2) \), so mode = 2.
(b)
The mean (expected value) is calculated as \( E(X) = \sum x P(X=x) \):
\( E(X) = 1(0.231) + 2(0.308) + 3(0.281) + 4(0.154) + 5(0.026) \)
\( = 0.231 + 0.616 + 0.843 + 0.616 + 0.13 = 2.436 \)
Mean \( \approx 2.44 \)
(c)
Selecting the first 50 customers based on ease of access is a convenience sampling method.