1. 1
   The random variable $X$ is normally distributed with mean 50 and standard deviation 4. A calculator may be used in part (c).

   (a)

   Find the $z$-value of $x=56$.

   [1 mark]

   • A$6$
   • B$1.5$
   • C$0.375$
   • D$-1.5$

   (b)

   Find the value of $x$ whose $z$-value is $-2.25$.

   [1 mark]

   • A$59$
   • B$47.75$
   • C$41$
   • D$14$

   (c)

   By standardising, find $P(X>56)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Amira and Ben took different tests. Marks on Amira's history test were normally distributed with mean 60 and standard deviation 8, and Amira scored 72. Marks on Ben's mathematics test were normally distributed with mean 70 and standard deviation 5, and Ben scored 81.

   (a)

   Find the $z$-value of Amira's mark.

   [1 mark]

   • A$1.5$
   • B$12$
   • C$0.1875$
   • D$2.2$

   (b)

   Which statement is correct?

   [1 mark]

   • AAmira's mark is 1.5 standard deviations below the mean of her test
   • BBoth marks are the same number of standard deviations above their means
   • CAmira did better relative to her group, because her test had the larger standard deviation
   • DRelative to their groups, Ben did better, because his mark is 2.2 standard deviations above his mean

   (c)

   Find the mark on Ben's test that would represent the same relative performance as Amira's mark.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The times taken by competitors to complete a puzzle are normally distributed with mean $\mu$ seconds and standard deviation 6 seconds. It is known that 20% of competitors complete the puzzle in less than 40 seconds. A calculator may be used in this question.

   (a)

   Find the value of $\mu$.

   [3 marks]

   (b)

   150 competitors enter a competition. Find the expected number of competitors who take more than 55 seconds to complete the puzzle.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory makes bolts whose lengths, $L$ mm, are normally distributed with mean $\mu$ mm and standard deviation $\sigma$ mm. It is found that 5% of the bolts are shorter than 49.2 mm and 10% of the bolts are longer than 50.9 mm. A calculator may be used in this question.

   (a)

   Find the value of $\mu$ and the value of $\sigma$.

   [6 marks]

   (b)

   A bolt is rejected if its length differs from $\mu$ by more than $d$ mm. The factory wants exactly 3% of bolts to be rejected.
   (i) Find the value of $d$.
   (ii) The bolts are packed in boxes of 20. Assuming that 3% of the bolts are rejected, independently, find the probability that a box contains at most one bolt that would be rejected.

   [6 marks]

   Total for question 4: 12 marks

End of questions