Home / IB Mathematics SL 4.9 The normal distribution -AI SL Paper 1- Exam Style Questions

IB Mathematics SL 4.9 The normal distribution -AI SL Paper 1- Exam Style Questions- New Syllabus

IB DP Maths AI SL Paper 1 - All Topics

Question 

Emma manages a factory that produces bags of sugar with a labelled weight of 500 g. The weights of the bags are normally distributed with a mean of 500 g and a standard deviation of 3 g.
(a) Write down the percentage of bags that weigh more than 500 g. [1]
A bag that weighs less than 495 g is rejected by the factory for being underweight.
(b) Find the probability that a randomly chosen bag is rejected for being underweight. [2]
A bag that weighs more than \( k \) grams is rejected by the factory for being overweight. The factory rejects 2% of bags for being overweight.
(c) Find the value of \( k \). [3]
▶️ Answer/Explanation
Markscheme
(a)
For a normal distribution with mean 500 g, the percentage of bags weighing more than 500 g is:
\( P(X > 500) = 50\% \). A1
[1 mark]
(b)
Calculate the z-score for \( X = 495 \):
\[ \begin{aligned} z &= \frac{495 – 500}{3} = \frac{-5}{3} \approx -1.6667 \end{aligned} \]
Find the probability:
\( P(X < 495) = P(Z < -1.6667) \approx 0.0478 \). A2
[2 marks]
(c)
Given \( P(X > k) = 0.02 \), then \( P(X < k) = 0.98 \). M1
Find the z-score for \( P(Z < z) = 0.98 \): \( z \approx 2.0537 \).
Solve for \( k \):
\[ \begin{aligned} k &= \mu + z \times \sigma \\ &= 500 + 2.0537 \times 3 \approx 506.161 \end{aligned} \] A2
Value of \( k \approx 506 \) g.
[3 marks]
Total Marks: 6
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